### The Mizar article:

### Series

**by****Konrad Raczkowski, and****Andrzej Nedzusiak**

- Received October 15, 1990
Copyright (c) 1990 Association of Mizar Users

- MML identifier: SERIES_1
- [ MML identifier index ]

environ vocabulary ARYTM, SEQ_1, FUNCT_1, POWER, SEQ_2, ORDINAL2, ARYTM_3, ARYTM_1, ABSVALUE, INT_1, SEQM_3, SUPINF_2, RLVECT_1, PROB_1, LATTICES, PREPOWER, SERIES_1, GROUP_1; notation ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, NAT_1, ABSVALUE, SEQ_1, SEQ_2, FUNCT_2, SEQM_3, INT_1, PREPOWER, POWER; constructors REAL_1, NAT_1, SEQ_2, SEQM_3, PREPOWER, POWER, PARTFUN1, MEMBERED, XBOOLE_0; clusters INT_1, XREAL_0, SEQ_1, NEWTON, RELSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements REAL, NUMERALS, SUBSET, ARITHM; theorems AXIOMS, SEQ_1, SEQ_2, SEQM_3, SEQ_4, REAL_1, REAL_2, SQUARE_1, NAT_1, RFUNCT_2, ABSVALUE, INT_1, PREPOWER, POWER, SCHEME1, FUNCT_2, NEWTON, XCMPLX_0, XCMPLX_1; schemes NAT_1, SEQ_1, RECDEF_1, RCOMP_1; begin reserve n,m,k for Nat; reserve a,p,r for real number; reserve s,s1,s2,s3 for Real_Sequence; Lm1: 1/1=1 & 1/(-1)=-1; theorem Th1: 0<a & a<1 & (for n holds s.n=a to_power (n+1)) implies s is convergent & lim s = 0 proof assume that A1: 0<a & a<1 and A2: for n holds s.n=a to_power (n+1); set r = 1/a - 1; 1/a > 1 by A1,Lm1,REAL_2:151; then A3: r > 0 by SQUARE_1:11; then A4: -1 < r by AXIOMS:22; A5: for p be real number st 0 < p ex m st for n st m<=n holds abs(s.n - 0) < p proof let p be real number; assume A6: 0<p; then A7: 1/p > 0 by REAL_2:127; A8: 1/(p * r) = 1/p/r by XCMPLX_1:79; [\1/(p * r)/] is Nat proof p * r > 0 by A3,A6,REAL_2:122; then A9: 1/(p * r) > 0 by REAL_2:127; 1/(p * r) <= [\1/(p * r)/] + 1 by INT_1:52; then 0 < [\1/(p * r)/] + 1 by A9,AXIOMS:22; then 0 <= [\1/(p * r)/] by INT_1:20; hence [\1/(p * r)/] is Nat by INT_1:16; end; then consider m such that A10: m = [\1/(p * r)/]; take m; now let n such that A11: m<=n; [\1/(p * r)/] > 1/(p * r) -1 by INT_1:def 4; then n > 1/(p * r) - 1 by A10,A11,AXIOMS:22; then n + 1 > 1/p/r by A8,REAL_1:84; then A12: (n+1) * r > 1/p by A3,REAL_2:177; a to_power (n+1) > 0 by A1,POWER:39; then A13: abs(a to_power (n+1)) = a to_power (n+1) by ABSVALUE:def 1; A14: 1 + (n+1) * r <= (1 + r) to_power (n + 1) by A4,POWER:56; 1 + r = 1/a by XCMPLX_1:27; then A15: (1+r) to_power (n+1) = 1 to_power (n+1) / a to_power (n+1) by A1, POWER:36 .= 1 / a to_power (n+1) by POWER:31; 0 + (n+1) * r < 1 + (n+1) * r by REAL_1:53; then 1/p < 1 + (n+1) * r by A12,AXIOMS:22; then 1/p < 1/ a to_power (n+1) by A14,A15,AXIOMS:22; then abs(a to_power (n+1)) < p by A7,A13,REAL_2:154; hence abs(s.n - 0) < p by A2; end; hence thesis; end; hence s is convergent by SEQ_2:def 6; hence lim s = 0 by A5,SEQ_2:def 7; end; theorem Th2: a <> 0 implies abs(a) to_power n = abs(a to_power n) proof assume A1: a <> 0; then A2: abs(a) > 0 by ABSVALUE:6; now per cases by A1,REAL_1:def 5; suppose A3: a>0; then A4: a to_power n = abs(a) to_power n by ABSVALUE:def 1; a to_power n > 0 by A3,POWER:39; hence abs(a) to_power n = abs(a to_power n) by A4,ABSVALUE:def 1; suppose A5: a < 0; A6: ex k st n = 2 * k or n = 2 * k + 1 by SCHEME1:1; reconsider m=n as Integer; now per cases by A6; suppose A7: (ex k being Integer st m = 2 * k); A8: abs(a) to_power n = (-a) to_power m by A5,ABSVALUE:def 1 .= a to_power m by A1,A7,POWER:54; abs(a) to_power n > 0 by A2,POWER:39; hence abs(a) to_power n = abs(a to_power n ) by A8,ABSVALUE:def 1; suppose A9: (ex k being Integer st m = 2 * k + 1); A10: abs(a) to_power n = (-a) to_power m by A5,ABSVALUE:def 1 .= - a to_power m by A1,A9,POWER:55; abs(a) to_power n > 0 by A2,POWER:39; hence abs(a) to_power n = abs( - a to_power n ) by A10,ABSVALUE:def 1 .= abs( a to_power n ) by ABSVALUE:17; end; hence abs(a) to_power n = abs(a to_power n); end; hence thesis; end; theorem Th3: abs(a)<1 & (for n holds s.n=a to_power (n+1)) implies s is convergent & lim s = 0 proof assume that A1: abs(a) < 1 and A2: for n holds s.n=a to_power (n+1); now per cases; suppose abs(a) = 0; then A3: a = 0 by ABSVALUE:7; A4: now let n; n+1<>0 by NAT_1:21; then n+1>0 by NAT_1:19; then a to_power (n+1) = 0 by A3,POWER:def 2; hence s.n = 0 by A2; end; then A5: s is constant by SEQM_3:def 6; s.0 = 0 by A4; hence s is convergent & lim s = 0 by A5,SEQ_4:39,41; suppose A6:not abs(a) = 0; then A7: a <> 0 by ABSVALUE:7; abs(a) >= 0 by ABSVALUE:5; then A8: abs(a) > 0 by A6,REAL_1:def 5; deffunc U(Nat) = abs(a) to_power ($1+1); consider s1 such that A9: for n holds s1.n = U(n) from ExRealSeq; A10: s1 is convergent & lim s1 = 0 by A1,A8,A9,Th1; now let n; thus s1.n = abs(a) to_power (n+1) by A9 .= abs(a to_power (n+1)) by A7,Th2 .= abs(s.n) by A2; end; then abs(s) is convergent & lim abs(s) = 0 by A10,SEQ_1:16; hence s is convergent & lim s = 0 by SEQ_4:28; end; hence thesis; end; definition let s; func Partial_Sums(s) -> Real_Sequence means :Def1: it.0=s.0 & for n holds it.(n+1) = it.n + s.(n+1); existence proof deffunc U(Nat,Real) = $2 + s.($1+1); consider f being Function of NAT,REAL such that A1: f.0 = s.0 & for n being Nat holds f.(n+1) = U(n,f.n) from LambdaRecExD; reconsider f as Real_Sequence; take f; thus thesis by A1; end; uniqueness proof let s1,s2; assume that A2: s1.0=s.0 & for n holds s1.(n+1)=s1.n + s.(n+1) and A3: s2.0=s.0 & for n holds s2.(n+1)=s2.n + s.(n+1); defpred X[Nat] means s1.$1 = s2.$1; A4: X[0] by A2,A3; A5: for k st X[k] holds X[k+1] proof let k; assume s1.k =s2.k; hence s1.(k+1) = s2.k + s.(k+1) by A2 .= s2.(k+1) by A3; end; for n holds X[n] from Ind(A4,A5); hence s1 = s2 by FUNCT_2:113; end; end; definition let s; attr s is summable means :Def2: Partial_Sums(s) is convergent; func Sum(s) -> Real equals :Def3: lim Partial_Sums(s); correctness; end; canceled 3; theorem Th7: s is summable implies s is convergent & lim s = 0 proof assume s is summable; then A1: Partial_Sums(s) is convergent by Def2; then A2: Partial_Sums(s) ^\1 is convergent & lim(Partial_Sums(s) ^\1) = lim(Partial_Sums(s)) by SEQ_4:33; then A3: lim(Partial_Sums(s) ^\1 - Partial_Sums(s)) = lim(Partial_Sums(s)) - lim(Partial_Sums(s)) by A1,SEQ_2:26 .= 0 by XCMPLX_1:14; A4: s ^\1 = Partial_Sums(s) ^\1 - Partial_Sums(s) proof now let n; (Partial_Sums(s)).(n+1) = (Partial_Sums(s)).n + s.(n+1) by Def1; then (Partial_Sums(s)).(n+1) = (Partial_Sums(s)).n + (s ^\1).n by SEQM_3:def 9; hence (Partial_Sums(s) ^\1).n = (Partial_Sums(s)).n + (s ^\1).n by SEQM_3:def 9; end; then A5: (Partial_Sums(s) ^\1) = Partial_Sums(s) + s ^\1 by SEQ_1:11; s ^\1 + (Partial_Sums(s) - Partial_Sums(s)) = s ^\1 proof now let n; thus (s ^\1 + (Partial_Sums(s) - Partial_Sums(s))).n = (s ^\1).n + (Partial_Sums(s) - Partial_Sums(s)).n by SEQ_1:11 .= (s ^\1).n + (Partial_Sums(s) +- Partial_Sums(s)).n by SEQ_1:15 .= (s ^\1).n + ((Partial_Sums(s)).n + (-Partial_Sums(s)).n) by SEQ_1:11 .= (s ^\1).n + ((Partial_Sums(s)).n +- (Partial_Sums(s)).n) by SEQ_1:14 .= (s ^\1).n + ((Partial_Sums(s)).n - (Partial_Sums(s)).n) by XCMPLX_0: def 8 .= (s ^\1).n + 0 by XCMPLX_1:14 .= (s ^\1).n; end; hence thesis by FUNCT_2:113; end; hence thesis by A5,SEQ_1:39; end; then A6: s ^\1 is convergent by A1,A2,SEQ_2:25; hence s is convergent by SEQ_4:35; thus thesis by A3,A4,A6,SEQ_4:36; end; theorem Th8: Partial_Sums(s1) + Partial_Sums(s2) = Partial_Sums(s1+s2) proof A1: (Partial_Sums(s1) + Partial_Sums(s2)).0 = Partial_Sums(s1).0 + Partial_Sums(s2).0 by SEQ_1:11 .= s1.0 + Partial_Sums(s2).0 by Def1 .= s1.0 + s2.0 by Def1 .= (s1+s2).0 by SEQ_1:11; now let n; thus (Partial_Sums(s1) + Partial_Sums(s2)).(n+1) = Partial_Sums(s1).(n+1) + Partial_Sums(s2).(n+1) by SEQ_1:11 .= Partial_Sums(s1).n + s1.(n+1) + Partial_Sums(s2).(n+1) by Def1 .= Partial_Sums(s1).n+s1.(n+1)+(s2.(n+1)+Partial_Sums(s2).n) by Def1 .= Partial_Sums(s1).n+s1.(n+1)+s2.(n+1)+Partial_Sums(s2).n by XCMPLX_1:1 .= Partial_Sums(s1).n+(s1.(n+1)+s2.(n+1))+Partial_Sums(s2).n by XCMPLX_1:1 .= Partial_Sums(s1).n+(s1+s2).(n+1)+Partial_Sums(s2).n by SEQ_1:11 .= Partial_Sums(s1).n+Partial_Sums(s2).n+(s1+s2).(n+1) by XCMPLX_1:1 .= (Partial_Sums(s1)+Partial_Sums(s2)).n+(s1+s2).(n+1) by SEQ_1:11; end; hence thesis by A1,Def1; end; theorem Th9: Partial_Sums(s1) - Partial_Sums(s2) = Partial_Sums(s1-s2) proof A1: (Partial_Sums(s1) - Partial_Sums(s2)).0 = Partial_Sums(s1).0 - Partial_Sums(s2).0 by RFUNCT_2:6 .= s1.0 - Partial_Sums(s2).0 by Def1 .= s1.0 - s2.0 by Def1 .= (s1-s2).0 by RFUNCT_2:6; now let n; thus (Partial_Sums(s1) - Partial_Sums(s2)).(n+1) = Partial_Sums(s1).(n+1) - Partial_Sums(s2).(n+1) by RFUNCT_2:6 .= (Partial_Sums(s1).n + s1.(n+1)) - Partial_Sums(s2).(n+1) by Def1 .= (Partial_Sums(s1).n+s1.(n+1))-(s2.(n+1)+Partial_Sums(s2).n) by Def1 .= (Partial_Sums(s1).n+s1.(n+1))-s2.(n+1)-Partial_Sums(s2).n by XCMPLX_1:36 .= Partial_Sums(s1).n+(s1.(n+1)-s2.(n+1))-Partial_Sums(s2).n by XCMPLX_1:29 .= (s1-s2).(n+1)+Partial_Sums(s1).n-Partial_Sums(s2).n by RFUNCT_2:6 .= (s1-s2).(n+1)+(Partial_Sums(s1).n-Partial_Sums(s2).n) by XCMPLX_1:29 .= (Partial_Sums(s1)-Partial_Sums(s2)).n+(s1-s2).(n+1) by RFUNCT_2:6; end; hence thesis by A1,Def1; end; theorem s1 is summable & s2 is summable implies s1+s2 is summable & Sum(s1+s2) = Sum(s1) + Sum(s2) proof assume A1: s1 is summable & s2 is summable; then A2: Partial_Sums(s1) is convergent by Def2; A3: Partial_Sums(s2) is convergent by A1,Def2; then Partial_Sums(s1) + Partial_Sums(s2) is convergent by A2,SEQ_2:19; then Partial_Sums(s1+s2) is convergent by Th8; hence s1+s2 is summable by Def2; thus Sum(s1+s2) =lim Partial_Sums(s1+s2) by Def3 .=lim (Partial_Sums(s1) + Partial_Sums(s2)) by Th8 .=lim Partial_Sums(s1) + lim Partial_Sums(s2) by A2,A3,SEQ_2:20 .=Sum(s1) + lim Partial_Sums(s2) by Def3 .=Sum(s1) + Sum(s2) by Def3; end; theorem s1 is summable & s2 is summable implies s1-s2 is summable & Sum(s1-s2) = Sum(s1) - Sum(s2) proof assume A1: s1 is summable & s2 is summable; then A2: Partial_Sums(s1) is convergent by Def2; A3: Partial_Sums(s2) is convergent by A1,Def2; then Partial_Sums(s1) - Partial_Sums(s2) is convergent by A2,SEQ_2:25; then Partial_Sums(s1-s2) is convergent by Th9; hence s1-s2 is summable by Def2; thus Sum(s1-s2) =lim Partial_Sums(s1-s2) by Def3 .=lim (Partial_Sums(s1) - Partial_Sums(s2)) by Th9 .=lim Partial_Sums(s1) - lim Partial_Sums(s2) by A2,A3,SEQ_2:26 .=Sum(s1) - lim Partial_Sums(s2) by Def3 .=Sum(s1) - Sum(s2) by Def3; end; theorem Th12: Partial_Sums(r(#)s) = r(#)Partial_Sums(s) proof A1: (r(#)Partial_Sums(s)).0 = r*Partial_Sums(s).0 by SEQ_1:13 .= r*s.0 by Def1 .= (r(#)s).0 by SEQ_1:13; now let n; thus (r(#)Partial_Sums(s)).(n+1) = r*Partial_Sums(s).(n+1) by SEQ_1:13 .= r*(Partial_Sums(s).n + s.(n+1)) by Def1 .= r*Partial_Sums(s).n + r*s.(n+1) by XCMPLX_1:8 .= (r(#)Partial_Sums(s)).n + r*s.(n+1) by SEQ_1:13 .= (r(#)Partial_Sums(s)).n + (r(#)s).(n+1) by SEQ_1:13; end; hence thesis by A1,Def1; end; theorem Th13: s is summable implies r(#)s is summable & Sum(r(#)s) =r*Sum(s) proof assume s is summable; then A1: Partial_Sums(s) is convergent by Def2; then r(#)Partial_Sums(s) is convergent by SEQ_2:21; then Partial_Sums(r(#)s) is convergent by Th12; hence r(#)s is summable by Def2; thus Sum(r(#)s) =lim Partial_Sums(r(#)s) by Def3 .=lim (r(#)Partial_Sums(s)) by Th12 .=r*(lim Partial_Sums(s)) by A1,SEQ_2:22 .=r*Sum(s) by Def3; end; theorem Th14: for s,s1 st for n holds s1.n=s.0 holds Partial_Sums(s^\1) = (Partial_Sums(s)^\1) - s1 proof let s,s1; assume A1: for n holds s1.n=s.0; A2: ((Partial_Sums(s)^\1) - s1).0 = (Partial_Sums(s)^\1).0 - s1.0 by RFUNCT_2:6 .= (Partial_Sums(s)^\1).0 - s.0 by A1 .= Partial_Sums(s).(0+1) - s.0 by SEQM_3:def 9 .= Partial_Sums(s).0 + s.(0+1) - s.0 by Def1 .= s.(0+1) + s.0 - s.0 by Def1 .= s.(0+1) + (s.0 - s.0) by XCMPLX_1:29 .= s.(0+1) + 0 by XCMPLX_1:14 .= (s^\1).0 by SEQM_3:def 9; now let k; thus ((Partial_Sums(s)^\1) - s1).(k+1) = (Partial_Sums(s)^\1).(k+1) - s1.(k+1) by RFUNCT_2:6 .= (Partial_Sums(s)^\1).(k+1) - s.0 by A1 .= Partial_Sums(s).(k+1+1) - s.0 by SEQM_3:def 9 .= s.(k+1+1) + Partial_Sums(s).(k+1) - s.0 by Def1 .= s.(k+1+1) + Partial_Sums(s).(k+1) - s1.k by A1 .= s.(k+1+1) + (Partial_Sums(s).(k+1) - s1.k) by XCMPLX_1:29 .= s.(k+1+1) + ((Partial_Sums(s)^\1).k - s1.k) by SEQM_3:def 9 .= s.(k+1+1) + ((Partial_Sums(s)^\1) - s1).k by RFUNCT_2:6 .= ((Partial_Sums(s)^\1) - s1).k + (s^\1).(k+1) by SEQM_3:def 9; end; hence thesis by A2,Def1; end; theorem Th15: s is summable implies for n holds s^\n is summable proof defpred X[Nat] means s^\$1 is summable; assume s is summable; then A1: X[0] by SEQM_3:34; A2: for n st X[n] holds X[n+1] proof let n; assume s^\n is summable; then Partial_Sums(s^\n) is convergent by Def2; then A3: Partial_Sums(s^\n)^\1 is convergent by SEQ_4:33; A4: s^\(n+1)=(s^\n)^\1 by SEQM_3:36; deffunc U(Nat) = (s^\n).0; consider s1 such that A5: for k holds s1.k = U(k) from ExRealSeq; s1 is constant by A5,SEQM_3:def 6; then A6: s1 is convergent by SEQ_4:39; Partial_Sums(s^\n^\1) = (Partial_Sums(s^\n)^\1) - s1 by A5,Th14; then Partial_Sums(s^\n^\1) is convergent by A3,A6,SEQ_2:25; hence thesis by A4,Def2; end; thus for n holds X[n] from Ind(A1,A2); end; theorem Th16: (ex n st s^\n is summable) implies s is summable proof given n such that A1: s^\n is summable; s^\n^\1 is summable by A1,Th15; then A2: Partial_Sums(s^\n^\1) is convergent by Def2; deffunc U(Nat) = Partial_Sums(s).n; consider s1 such that A3: for k holds s1.k= U(k) from ExRealSeq; s1 is constant by A3,SEQM_3:def 6; then A4: s1 is convergent by SEQ_4:39; for k holds (Partial_Sums(s)^\(n+1)).k = Partial_Sums(s^\(n+1)).k + s1.k proof A5: Partial_Sums(s^\(n+1)).0 = (s^\(n+1)).0 by Def1 .= s.(0+(n+1)) by SEQM_3:def 9 .= s.(n+1); defpred X[Nat] means (Partial_Sums(s)^\(n+1)).$1 = Partial_Sums(s^\(n+1)).$1 + s1.$1; (Partial_Sums(s)^\(n+1)).0 = Partial_Sums(s).(0+(n+1)) by SEQM_3:def 9 .= s.(n+1) + Partial_Sums(s).n by Def1 .= Partial_Sums(s^\(n+1)).0 + s1.0 by A3,A5; then A6: X[0]; A7: for k st X[k] holds X[k+1] proof let k; assume A8: (Partial_Sums(s)^\(n+1)).k = Partial_Sums(s^\(n+1)).k + s1.k; Partial_Sums(s^\(n+1)).(k+1) + s1.(k+1) = s1.(k+1) + (Partial_Sums(s^\(n+1)).k + (s^\(n+1)).(k+1)) by Def1 .= s1.(k+1) + Partial_Sums(s^\(n+1)).k + (s^\(n+1)).(k+1) by XCMPLX_1:1 .= Partial_Sums(s).n + Partial_Sums(s^\(n+1)).k + (s^\(n+1)).(k+1) by A3 .= (Partial_Sums(s)^\(n+1)).k + (s^\(n+1)).(k+1) by A3,A8 .= Partial_Sums(s).(k+(n+1)) + (s^\(n+1)).(k+1) by SEQM_3:def 9 .= Partial_Sums(s).(k+(n+1)) + s.(k+1+(n+1)) by SEQM_3:def 9 .= Partial_Sums(s).(k+(n+1)) + s.(k+(n+1)+1) by XCMPLX_1:1 .= Partial_Sums(s).(k+(n+1)+1) by Def1 .= Partial_Sums(s).(k+1+(n+1)) by XCMPLX_1:1 .= (Partial_Sums(s)^\(n+1)).(k+1) by SEQM_3:def 9; hence (Partial_Sums(s)^\(n+1)).(k+1) = Partial_Sums(s^\(n+1)).(k+1) + s1.(k+1); end; thus for k holds X[k] from Ind(A6,A7); end; then Partial_Sums(s)^\(n+1) = Partial_Sums(s^\(n+1)) + s1 by SEQ_1:11 .= Partial_Sums((s^\n)^\1) + s1 by SEQM_3:36; then Partial_Sums(s)^\(n+1) is convergent by A2,A4,SEQ_2:19; then Partial_Sums(s) is convergent by SEQ_4:35; hence thesis by Def2; end; theorem Th17: (for n holds s1.n<=s2.n) implies for n holds Partial_Sums(s1).n<=Partial_Sums(s2).n proof assume A1: for n holds s1.n<=s2.n; A2: Partial_Sums(s2).0 = s2.0 by Def1; defpred X[Nat] means Partial_Sums(s1).$1 <= Partial_Sums(s2).$1; Partial_Sums(s1).0 = s1.0 by Def1; then A3: X[0] by A1,A2; A4: for n st X[n] holds X[n+1] proof let n such that A5: Partial_Sums(s1).n <= Partial_Sums(s2).n; A6: Partial_Sums(s1).(n+1) = Partial_Sums(s1).n + s1.(n+1) by Def1; A7: Partial_Sums(s2).(n+1) = Partial_Sums(s2).n + s2.(n+1) by Def1; s1.(n+1)<=s2.(n+1) by A1; hence Partial_Sums(s1).(n+1) <= Partial_Sums(s2).(n+1) by A5,A6,A7,REAL_1:55; end; thus for n holds X[n] from Ind(A3,A4); end; theorem s is summable implies for n holds Sum(s) = Partial_Sums(s).n + Sum(s^\(n+1)) proof assume A1: s is summable; defpred X[Nat] means Sum(s) = Partial_Sums(s).$1 + Sum(s^\($1+1)); A2: X[0] proof deffunc U(Nat) = s.0; consider s1 such that A3: for k holds s1.k=U(k) from ExRealSeq; A4: Partial_Sums(s) is convergent by A1,Def2; then A5: Partial_Sums(s)^\1 is convergent by SEQ_4:33; A6: s1 is constant by A3,SEQM_3:def 6; then A7: s1 is convergent by SEQ_4:39; Partial_Sums(s^\1) = (Partial_Sums(s)^\1) - s1 by A3,Th14; then lim Partial_Sums(s^\1) = lim (Partial_Sums(s)^\1) - lim s1 by A5,A7, SEQ_2:26 .= lim Partial_Sums(s) - lim s1 by A4,SEQ_4:33 .= Sum(s) - lim s1 by Def3 .= Sum(s) - s1.0 by A6,SEQ_4:41 .= Sum(s) - s.0 by A3; then Sum(s^\1) = Sum(s) - s.0 by Def3; then Sum(s^\1) = Sum(s) +- s.0 by XCMPLX_0:def 8; then Sum(s) = Sum(s^\1) -(-s.0) by XCMPLX_1:26; then Sum(s) = Sum(s^\1) +(-(-s.0)) by XCMPLX_0:def 8; hence thesis by Def1; end; A8: for n st X[n] holds X[n+1] proof let n; assume A9: Sum(s) = Partial_Sums(s).n + Sum(s^\(n+1)); deffunc U(Nat) = (s^\(n+1)).0; consider s1 such that A10: for k holds s1.k=U(k) from ExRealSeq; A11: s1 is constant by A10,SEQM_3:def 6; then A12: s1 is convergent by SEQ_4:39; s^\(n+1) is summable by A1,Th15; then A13: Partial_Sums(s^\(n+1)) is convergent by Def2; then A14: Partial_Sums(s^\(n+1))^\1 is convergent by SEQ_4:33; A15: Partial_Sums(s^\(n+1)^\1) = (Partial_Sums(s^\(n+1))^\1) - s1 by A10,Th14 ; lim Partial_Sums(s^\(n+1+1)) = lim Partial_Sums(s^\(n+1)^\1) by SEQM_3:36 .= lim (Partial_Sums(s^\(n+1))^\1) - lim s1 by A12,A14,A15,SEQ_2:26 .= lim Partial_Sums(s^\(n+1)) - lim s1 by A13,SEQ_4:33 .= Sum(s^\(n+1)) - lim s1 by Def3 .= Sum(s^\(n+1)) - s1.0 by A11,SEQ_4:41 .= Sum(s^\(n+1)) - (s^\(n+1)).0 by A10; then Sum(s^\(n+1+1)) = Sum(s^\(n+1)) - (s^\(n+1)).0 by Def3 .= Sum(s) - Partial_Sums(s).n - (s^\(n+1)).0 by A9,XCMPLX_1:26 .= Sum(s) - (Partial_Sums(s).n + (s^\(n+1)).0) by XCMPLX_1:36 .= Sum(s) - (Partial_Sums(s).n + s.(0+(n+1))) by SEQM_3:def 9 .= Sum(s) - Partial_Sums(s).(n+1) by Def1; then Sum(s^\(n+1+1)) = Sum(s) +- Partial_Sums(s).(n+1) by XCMPLX_0:def 8; hence Sum(s) = Sum(s^\(n+1+1)) -(-Partial_Sums(s).(n+1)) by XCMPLX_1:26 .= Sum(s^\(n+1+1)) +(-(-Partial_Sums(s).(n+1))) by XCMPLX_0:def 8 .= Partial_Sums(s).(n+1) + Sum(s^\(n+1+1)); end; thus for n holds X[n] from Ind(A2,A8); end; theorem Th19: (for n holds 0<=s.n) implies Partial_Sums(s) is non-decreasing proof assume A1: for n holds 0<=s.n; now let n; 0<=s.(n+1) by A1; then 0 + Partial_Sums(s).n <= s.(n+1) + Partial_Sums(s).n by AXIOMS:24;hence Partial_Sums(s).n <= Partial_Sums(s).(n+1) by Def1; end; hence thesis by SEQM_3:def 13; end; theorem Th20: (for n holds 0<=s.n) implies (Partial_Sums(s) is bounded_above iff s is summable) proof assume for n holds 0<=s.n; then A1: Partial_Sums(s) is non-decreasing by Th19; thus Partial_Sums(s) is bounded_above implies s is summable proof assume Partial_Sums(s) is bounded_above; then Partial_Sums(s) is convergent by A1,SEQ_4:51; hence thesis by Def2; end; assume s is summable; then Partial_Sums(s) is convergent by Def2; then Partial_Sums(s) is bounded by SEQ_2:27; hence thesis by SEQ_2:def 5; end; theorem s is summable & (for n holds 0<=s.n) implies 0<=Sum(s) proof assume that A1: s is summable and A2: for n holds 0<=s.n; A3: Partial_Sums(s) is non-decreasing by A2,Th19; A4: Partial_Sums(s) is convergent by A1,Def2; now let n; A5: Partial_Sums(s).0<=Partial_Sums(s).n by A3,SEQM_3:21; 0<=s.0 by A2; then 0<=Partial_Sums(s).0 by Def1; hence 0<=Partial_Sums(s).n by A5,AXIOMS:22; end; then 0 <= lim Partial_Sums(s) by A4,SEQ_2:31; hence thesis by Def3; end; theorem Th22: (for n holds 0<=s2.n) & s1 is summable & (ex m st for n st m<=n holds s2.n<=s1.n) implies s2 is summable proof assume that A1: for n holds 0<=s2.n and A2: s1 is summable; given m such that A3: for n st m<=n holds s2.n<=s1.n; A4: now let n; A5: (s1^\m).n = s1.(n+m) by SEQM_3:def 9; A6: 0<= s2.(n+m) by A1; n+m>=m by NAT_1:29; then s2.(n+m)<=s1.(n+m) by A3; hence 0<=(s1^\m).n by A5,A6,AXIOMS:22; end; s1^\m is summable by A2,Th15; then Partial_Sums(s1^\m) is bounded_above by A4,Th20; then consider r being real number such that A7: for n holds Partial_Sums(s1^\m).n<r by SEQ_2:def 3; A8: now let n; m<=n+m by NAT_1:37; then s2.(n+m)<=s1.(n+m) by A3; then (s2^\m).n<=s1.(n+m) by SEQM_3:def 9; hence (s2^\m).n<=(s1^\m).n by SEQM_3:def 9; end; A9: now let n; (s2^\m).n = s2.(n+m) by SEQM_3:def 9; hence 0<=(s2^\m).n by A1; end; now let n; A10: Partial_Sums(s1^\m).n<r by A7; Partial_Sums(s2^\m).n <= Partial_Sums(s1^\m).n by A8,Th17; hence Partial_Sums(s2^\m).n<r by A10,AXIOMS:22; end; then Partial_Sums(s2^\m) is bounded_above by SEQ_2:def 3; then s2^\m is summable by A9,Th20; hence thesis by Th16; end; canceled; theorem (for n holds 0<=s1.n & s1.n<=s2.n) & s2 is summable implies s1 is summable & Sum(s1)<=Sum(s2) proof assume that A1: for n holds 0<=s1.n & s1.n<=s2.n and A2: s2 is summable; for n holds 0<=n implies s1.n<=s2.n by A1; hence s1 is summable by A1,A2,Th22; then A3: Partial_Sums(s1) is convergent by Def2; A4: Partial_Sums(s2) is convergent by A2,Def2; for n holds Partial_Sums(s1).n<=Partial_Sums(s2).n by A1,Th17; then lim Partial_Sums(s1)<=lim Partial_Sums(s2) by A3,A4,SEQ_2:32; then Sum(s1)<=lim Partial_Sums(s2) by Def3; hence thesis by Def3; end; theorem Th25: s is summable iff for r st 0<r ex n st for m st n<=m holds abs(Partial_Sums(s).m - Partial_Sums(s).n)<r proof thus s is summable implies for r st 0<r ex n st for m st n<=m holds abs(Partial_Sums(s).m - Partial_Sums(s).n)<r proof assume s is summable; then Partial_Sums(s) is convergent by Def2; hence thesis by SEQ_4:58; end; assume for r st 0<r ex n st for m st n<=m holds abs(Partial_Sums(s).m - Partial_Sums(s).n)<r; then Partial_Sums(s) is convergent by SEQ_4:58; hence thesis by Def2; end; theorem Th26: a <> 1 implies Partial_Sums(a GeoSeq).n = (1 - a to_power (n+1))/(1-a) proof assume a<>1; then A1: 1-a <> 0 by XCMPLX_1:15; defpred X[Nat] means Partial_Sums(a GeoSeq).$1 = (1-a to_power ($1+1))/(1-a); Partial_Sums(a GeoSeq).0 = a GeoSeq.0 by Def1 .= 1 by PREPOWER:4 .= (1-a)/(1-a) by A1,XCMPLX_1:60 .= (1-a to_power (0+1))/(1-a) by POWER:30; then A2: X[0]; A3: for n st X[n] holds X[n+1] proof let n; assume A4: Partial_Sums(a GeoSeq).n = (1 - a to_power (n+1))/(1-a); 0<=n by NAT_1:18; then A5: 0+1<=n+1 by AXIOMS:24; n+1<=n+1+1 by REAL_1:69; then A6: 1<=n+1+1 by A5,AXIOMS:22; thus Partial_Sums(a GeoSeq).(n+1) = (1 - a to_power (n+1))/(1-a) + a GeoSeq.(n+1) by A4,Def1 .= (1 - a to_power (n+1))/(1-a) + a |^ (n+1) by PREPOWER:def 1 .= (1 - a to_power (n+1))/(1-a) + a to_power (n+1) * 1 by A5,POWER:47 .= (1-a to_power (n+1))/(1-a)+a to_power (n+1)*((1-a)/(1-a)) by A1,XCMPLX_1: 60 .= (1-a to_power (n+1))/(1-a)+(a to_power (n+1)*(1-a))/(1-a) by XCMPLX_1:75 .= (1-a to_power (n+1) + a to_power (n+1)*(1-a))/(1-a) by XCMPLX_1:63 .= (1-a to_power (n+1) + (a to_power (n+1)*1-a to_power (n+1)*a))/(1-a) by XCMPLX_1:40 .= (1-a to_power (n+1) + (a to_power (n+1)-a |^ (n+1)*a))/(1-a) by A5,POWER:47 .= (1-a to_power (n+1) + (a to_power (n+1)-a |^ (n+1+1)))/(1-a) by NEWTON:11 .= (1-a to_power (n+1) + (a to_power (n+1)-a to_power (n+1+1)))/(1-a) by A6,POWER:47 .= (1-a to_power (n+1) + a to_power (n+1) - a to_power (n+1+1))/(1-a) by XCMPLX_1:29 .= (1 - a to_power (n+1+1))/(1-a) by XCMPLX_1:27; end; for n holds X[n] from Ind(A2,A3); hence thesis; end; theorem a <> 1 & (for n holds s.(n+1) = a * s.n) implies for n holds Partial_Sums(s).n = s.0 * (1 - a to_power (n+1))/(1-a) proof assume A1: a <> 1 & (for n holds s.(n+1) = a * s.n); defpred X[Nat] means s.$1 = s.0 * a GeoSeq.$1; a GeoSeq.0 = 1 by PREPOWER:4; then A2: X[0]; A3: for n st X[n] holds X[n+1] proof let n; assume s.n = s.0 * a GeoSeq.n; then s.(n+1) = a * (s.0 * a GeoSeq.n) by A1 .= s.0 * (a GeoSeq.n * a) by XCMPLX_1:4 .= s.0 * a GeoSeq.(n+1) by PREPOWER:4; hence thesis; end; for n holds X[n] from Ind(A2,A3); then s = s.0 (#) a GeoSeq by SEQ_1:13; then A4: Partial_Sums(s) = s.0 (#) Partial_Sums(a GeoSeq) by Th12; now let n; thus Partial_Sums(s).n = s.0 * Partial_Sums(a GeoSeq).n by A4,SEQ_1:13 .= s.0 * ((1 - a to_power (n+1))/(1-a)) by A1,Th26 .= s.0 * (1 - a to_power (n+1))/(1-a) by XCMPLX_1:75; end; hence thesis; end; theorem Th28: abs(a)<1 implies a GeoSeq is summable & Sum(a GeoSeq) = 1/(1-a) proof assume A1: abs(a)<1; deffunc U(Nat) = a to_power ($1+1); consider s such that A2: for n holds s.n = U(n) from ExRealSeq; A3: s is convergent & lim s = 0 by A1,A2,Th3; A4: now let r be real number such that A5: r>0; A6: a<1 by A1,SEQ_2:9; then A7: 1-a>0 by SQUARE_1:11; then r*(1-a)>0*r by A5,REAL_1:70; then consider m such that A8: for n st n>=m holds abs(s.n - 0)<r*(1-a) by A3,SEQ_2:def 7; take m; A9: a<>1 by A1,SEQ_2:9; A10: 1-a<>0 by A6,SQUARE_1:11; let n such that A11: n>=m; A12: abs(Partial_Sums(a GeoSeq).n - 1/(1-a)) = abs((1 - a to_power (n+1))/(1-a) - 1/(1-a)) by A9,Th26 .= abs((1 - a to_power (n+1) - 1)/(1-a)) by XCMPLX_1:121 .= abs((1 +- a to_power (n+1) - 1)/(1-a)) by XCMPLX_0:def 8 .= abs((- a to_power (n+1))/(1-a)) by XCMPLX_1:26 .= abs(- a to_power (n+1)/(1-a)) by XCMPLX_1:188 .= abs(a to_power (n+1)/(1-a)) by ABSVALUE:17 .= abs(a to_power (n+1))/abs((1-a)) by ABSVALUE:16 .= abs(a to_power (n+1))/(1-a) by A7,ABSVALUE:def 1; abs(s.n - 0)<r*(1-a) by A8,A11; then abs(a to_power (n+1) - 0)<r*(1-a) by A2; then abs(a to_power (n+1))/(1-a)<r*(1-a)/(1-a) by A7,REAL_1:73;hence abs(Partial_Sums(a GeoSeq).n - 1/(1-a)) < r by A10,A12,XCMPLX_1:90; end; then A13: Partial_Sums(a GeoSeq) is convergent by SEQ_2:def 6; then A14: lim Partial_Sums(a GeoSeq) = 1/(1-a) by A4,SEQ_2:def 7; thus a GeoSeq is summable by A13,Def2; thus thesis by A14,Def3; end; theorem abs(a) < 1 & (for n holds s.(n+1) = a * s.n) implies s is summable & Sum(s) = s.0/(1-a) proof assume A1: abs(a) < 1 & (for n holds s.(n+1) = a * s.n); defpred X[Nat] means s.$1 = s.0 * a GeoSeq.$1; a GeoSeq.0 = 1 by PREPOWER:4; then A2: X[0]; A3: for n st X[n] holds X[n+1] proof let n; assume s.n = s.0 * a GeoSeq.n; then s.(n+1) = a * (s.0 * a GeoSeq.n) by A1 .= s.0 * (a GeoSeq.n * a) by XCMPLX_1:4 .= s.0 * a GeoSeq.(n+1) by PREPOWER:4; hence thesis; end; for n holds X[n] from Ind(A2,A3); then s = s.0 (#) a GeoSeq by SEQ_1:13; then A4: Partial_Sums(s) = s.0 (#) Partial_Sums(a GeoSeq) by Th12; A5: a GeoSeq is summable & Sum(a GeoSeq) = 1/(1-a) by A1,Th28; then A6: Partial_Sums(a GeoSeq) is convergent by Def2; then A7: Partial_Sums(s) is convergent by A4,SEQ_2:21; lim Partial_Sums(a GeoSeq) = 1/(1-a) by A5,Def3; then A8: lim Partial_Sums(s) = s.0 * (1/(1-a)) by A4,A6,SEQ_2:22 .= s.0*1/(1-a) by XCMPLX_1:75 .= s.0/(1-a); thus s is summable by A7,Def2; thus thesis by A8,Def3; end; theorem Th30: (for n holds s.n>0 & s1.n=s.(n+1)/s.n) & s1 is convergent & lim s1 < 1 implies s is summable proof assume that A1: (for n holds s.n>0 & s1.n=s.(n+1)/s.n) and A2: s1 is convergent and A3: lim s1 < 1; set r = (1 - lim s1)/2; 0 + lim s1 < 1 by A3; then 0 < 1 - lim s1 by REAL_1:86; then A4: r > 0 by SEQ_2:3; r * (1+1) = (1 - lim s1)/2 * 2; then r*1 + r = (1 - lim s1)/2 * 2 by XCMPLX_1:8; then r + r = 1 - lim s1 by XCMPLX_1:88; then A5: r + lim s1 = 1 - r by XCMPLX_1:35; consider m such that A6: for n st m <= n holds abs(s1.n - lim s1) < r by A2,A4,SEQ_2:def 7; set s2 = (s.m * (1-r) to_power (-m)) (#) (1-r) GeoSeq; A7: now let n; A8: s.n > 0 by A1; s.(n+1) > 0 by A1; then s.(n+1)/s.n > 0 by A8,SEQ_2:6;hence s1.n >= 0 by A1; end; then A9: lim s1 >= 0 by A2,PREPOWER:2; A10: s2 is summable proof 1 - lim s1 > 0 by A3,SQUARE_1:11; then r > 0 by SEQ_2:3; then A11: 1 - r < 1 - 0 by REAL_2:105; 1 - lim s1 <= 1 - 0 by A9,REAL_2:106; then 1 - lim s1 < 2 * 2 by AXIOMS:22; then r < 2 * 2 / 2 by REAL_1:73; then r < 1 + 1; then r - 1 < 1 by REAL_1:84; then - (r - 1) > - 1 by REAL_1:50; then 1 - r > -1 by XCMPLX_1:143; then abs(1-r) < 1 by A11,SEQ_2:9; then (1-r) GeoSeq is summable by Th28; hence thesis by Th13; end; A12: 1 - r > 0 proof -1 < lim s1 by A9,AXIOMS:22; then - lim s1 < 1 by REAL_2:109; then 1 + -lim s1 < 1 + 1 by REAL_1:53; then 1 - lim s1 < 2 by XCMPLX_0:def 8; then (1 - lim s1)/2 < 2/2 by REAL_1:73; hence thesis by SQUARE_1:11; end; defpred X[Nat] means s.(m+$1) <= s2.(m+$1); A13: X[0] proof s2.(m+0) = (s.m * (1-r) to_power (-m)) * (1-r) GeoSeq.m by SEQ_1:13 .= (s.m * (1-r) to_power (-m)) * (1-r) |^ m by PREPOWER:def 1 .= (s.m * (1-r) to_power (-m)) * (1-r) to_power m by A12,POWER:46 .= s.m * ((1-r) to_power (-m) * (1-r) to_power m) by XCMPLX_1:4 .= s.m * (1-r) to_power (-m + m) by A12,POWER:32 .= s.m * (1-r) to_power 0 by XCMPLX_0:def 6 .= s.m * 1 by POWER:29 .= s.(m+0); hence thesis; end; A14: for k holds X[k] implies X[k+1] proof let k such that A15: s.(m+k) <= s2.(m+k); A16: s1.(m+k) >= 0 by A7; s.(m+k) > 0 by A1; then A17: s2.(m+k) >= 0 by A15,AXIOMS:22; A18: s.(m+k) <> 0 by A1; A19: now m <= m+k by NAT_1:29; then abs(s1.(m+k) - lim s1) < r by A6; then s1.(m+k) - lim s1 < r by SEQ_2:9;hence s1.(m+k) <= 1 - r by A5,REAL_1:84; end; s.(m+(k+1)) = s.(m+k+1) by XCMPLX_1:1 .= s.(m+k+1) / s.(m+k) * s.(m+k) by A18,XCMPLX_1:88 .= s1.(m+k) * s.(m+k) by A1; then A20: s.(m+(k+1)) <= s1.(m+k) * s2.(m+k) by A15,A16,AXIOMS:25; A21: s1.(m+k) * s2.(m+k) <= (1-r) * s2.(m+k) by A17,A19,AXIOMS:25; set X = (s.m * (1-r) to_power (-m)); (1-r) * s2.(m+k) = (1-r) * (X * (1-r) GeoSeq.(m+k)) by SEQ_1:13 .= X * ((1-r) GeoSeq.(m+k) * (1-r)) by XCMPLX_1:4 .= X * (1-r) GeoSeq.(m+k+1) by PREPOWER:4 .= s2.(m+k+1) by SEQ_1:13 .= s2.(m+(k+1)) by XCMPLX_1:1; hence thesis by A20,A21,AXIOMS:22; end; A22: for k holds X[k] from Ind(A13,A14); A23: now let n; assume m <= n; then ex k st n = m + k by NAT_1:28; hence s.n <= s2.n by A22; end; for n holds 0 <= s.n by A1; hence thesis by A10,A23,Th22; end; theorem (for n holds s.n>0) & (ex m st for n st n>=m holds s.(n+1)/s.n >= 1) implies s is not summable proof assume that A1: (for n holds s.n>0) and A2: (ex m st for n st n>=m holds s.(n+1)/s.n >= 1); consider m such that A3: for n st n>=m holds s.(n+1)/s.n>=1 by A2; defpred X[Nat] means s.(m+$1)>=s.m; A4: X[0]; A5: for k holds X[k] implies X[k+1] proof let k such that A6: s.(m+k)>=s.m; A7: s.(m+k)>0 by A1; m+k>=m by NAT_1:29; then s.(m+k+1)/s.(m+k)>=1 by A3; then s.(m+k+1)>=s.(m+k) by A7,REAL_2:118; then s.(m+k+1)>=s.m by A6,AXIOMS:22; hence thesis by XCMPLX_1:1; end; A8: for k holds X[k] from Ind(A4,A5); A9: s.m>0 by A1; for k ex n st (n>=k & not abs(s.n - 0)<s.m) proof let k; take n = m + k; s.n>=s.m by A8; hence thesis by NAT_1:29,SEQ_2:9; end; then not lim s = 0 or s is not convergent by A9,SEQ_2:def 7; hence thesis by Th7; end; theorem Th32: (for n holds s.n>=0 & s1.n = n-root (s.n)) & s1 is convergent & lim s1 < 1 implies s is summable proof assume that A1: (for n holds s.n>=0 & s1.n = n-root (s.n)) and A2: s1 is convergent and A3: lim s1 < 1; set r = (1 - lim s1)/2; 0 + lim s1 < 1 by A3; then 0 < 1 - lim s1 by REAL_1:86; then A4: r > 0 by SEQ_2:3; r * (1+1) = (1 - lim s1)/2 * 2; then r*1 + r = (1 - lim s1)/2 * 2 by XCMPLX_1:8; then r + r = 1 - lim s1 by XCMPLX_1:88; then A5: r + lim s1 = 1 - r by XCMPLX_1:35; A6: (s1^\1) is convergent & lim (s1^\1) = lim s1 by A2,SEQ_4:33; now let n; A7: n+1>=1 by NAT_1:29; A8: s.(n+1) >= 0 by A1; (s1^\1).n = s1.(n+1) by SEQM_3:def 9 .= (n+1)-root (s.(n+1)) by A1; hence (s1^\1).n >= 0 by A7,A8,POWER:8; end; then A9: lim s1 >= 0 by A6,PREPOWER:2; A10: 1 - r > 0 proof -1 < lim s1 by A9,AXIOMS:22; then - lim s1 < 1 by REAL_2:109; then 1 + -lim s1 < 1 + 1 by REAL_1:53; then 1 - lim s1 < 2 by XCMPLX_0:def 8; then (1 - lim s1)/2 < 2/2 by REAL_1:73; hence thesis by SQUARE_1:11; end; consider m such that A11: for n st m <= n holds abs(s1.n - lim s1) < r by A2,A4,SEQ_2:def 7; A12: for n st m+1<=n holds s.n <= (1-r) to_power n proof let n; assume A13: m+1<=n; then A14: m<=n by NAT_1:38; 1<=m+1 by NAT_1:29; then A15: 1 <= n by A13,AXIOMS:22; then A16: 0 < n by AXIOMS:22; A17: s.n >= 0 by A1; abs(s1.n - lim s1) < r by A11,A14; then s1.n - lim s1 < r by SEQ_2:9; then s1.n < 1 - r by A5,REAL_1:84; then A18: n-root (s.n) < 1 - r by A1; now per cases; suppose s.n = 0; hence s.n < (1 - r) to_power n by A10,POWER:39; suppose s.n <> 0; then s.n > 0 by A17,REAL_1:def 5; then n -Root (s.n) > 0 by A15,PREPOWER:def 3; then n-root (s.n) > 0 by A15,A17,POWER:def 1; then (n-root (s.n)) to_power n < (1-r) to_power n by A16,A18,POWER:42; then (n-root (s.n)) |^ n < (1-r) to_power n by A15,POWER:47; hence s.n < (1-r) to_power n by A15,A17,POWER:5; end; hence thesis; end; set s2 = (1-r) GeoSeq; A19: s2 is summable proof 1 - lim s1 > 0 by A3,SQUARE_1:11; then r > 0 by SEQ_2:3; then A20: 1 - r < 1 - 0 by REAL_2:105; 1 - lim s1 <= 1 - 0 by A9,REAL_2:106; then 1 - lim s1 < 2 * 2 by AXIOMS:22; then r < 2 * 2 / 2 by REAL_1:73; then r < 1 + 1; then r - 1 < 1 by REAL_1:84; then - (r - 1) > - 1 by REAL_1:50; then 1 - r > -1 by XCMPLX_1:143; then abs(1-r) < 1 by A20,SEQ_2:9; hence thesis by Th28; end; for n st m+1 <= n holds s.n <= s2.n proof let n such that A21: m+1 <= n; 1<=m+1 by NAT_1:29; then A22: 1 <= n by A21,AXIOMS:22; s2.n = (1-r) |^ n by PREPOWER:def 1 .= (1-r) to_power n by A22,POWER:47; hence thesis by A12,A21; end; hence thesis by A1,A19,Th22; end; theorem Th33: (for n holds s.n>=0 & s1.n = n-root (s.n)) & (ex m st for n st m<=n holds s1.n>=1) implies s is not summable proof assume that A1:(for n holds s.n>=0 & s1.n = n-root (s.n)) and A2: (ex m st for n st m<=n holds s1.n>=1); consider m such that A3: for n st m<=n holds s1.n>=1 by A2; A4: for n st m+1<=n holds s.n>=1 proof let n such that A5: m+1<=n; m<=m+1 by NAT_1:29; then A6: m<=n by A5,AXIOMS:22; 1<=1+m by NAT_1:29; then A7: 1<=n by A5,AXIOMS:22; then A8: 0 < n by AXIOMS:22; s1.n >= 1 by A3,A6; then A9: n-root (s.n) >= 1 by A1; A10: s.n >= 0 by A1; now per cases by A9,REAL_1:def 5; suppose n-root (s.n) = 1; then s.n = 1 |^ n by A7,A10,POWER:5; hence s.n>=1 by NEWTON:15; suppose n-root (s.n) > 1; then (n-root (s.n)) to_power n > 1 by A8,POWER:40; then (n-root (s.n)) |^ n > 1 by A7,POWER:47; hence s.n >= 1 by A7,A10,POWER:5; end; hence thesis; end; for k ex n st k<=n & not abs(s.n - 0) < 1 proof let k; take n = m + 1 + k; n >= m + 1 by NAT_1:29; then not s.n < 1 by A4; hence thesis by NAT_1:29,SEQ_2:9; end; then s is not convergent or not lim s = 0 by SEQ_2:def 7; hence thesis by Th7; end; theorem (for n holds s.n>=0 & s1.n = n-root (s.n)) & s1 is convergent & lim s1 > 1 implies s is not summable proof assume that A1: (for n holds s.n>=0 & s1.n = n-root (s.n)) and A2: s1 is convergent & lim s1 > 1; set r = lim s1 - 1; A3: r > 0 by A2,SQUARE_1:11; A4: lim s1 - r = 1 by XCMPLX_1:18; consider m such that A5: for n st m<=n holds abs(s1.n - lim s1) < r by A2,A3,SEQ_2:def 7; for n st m<=n holds s1.n >= 1 proof let n; assume m<=n; then abs(s1.n - lim s1) < r by A5; then s1.n - lim s1 > - r by SEQ_2:9; then s1.n - lim s1 + lim s1 > - r + lim s1 by REAL_1:53; then s1.n > lim s1 + -r by XCMPLX_1:27; hence thesis by A4,XCMPLX_0:def 8; end; hence thesis by A1,Th33; end; definition let k, n be Nat; redefine func k to_power n -> Nat; coherence proof defpred X[Nat] means k to_power $1 is Nat; A1: X[0] by POWER:29; A2: for m st X[m] holds X[m+1] proof let m; assume k to_power m is Nat; then reconsider k1 = k to_power m as Nat; per cases by NAT_1:19; suppose A3: k = 0; m >= 0 by NAT_1:18; then m+1 > 0 by NAT_1:38; hence thesis by A3,POWER:def 2; suppose k > 0; then k to_power (m+1) = k to_power m * k to_power 1 by POWER:32 .= k1 * k by POWER:30; hence thesis; end; for m holds X[m] from Ind(A1,A2); hence k to_power n is Nat; end; end; theorem Th35: s is non-increasing & (for n holds s.n>=0 & s1.n = 2 to_power n * s.(2 to_power n)) implies (s is summable iff s1 is summable) proof assume that A1: s is non-increasing and A2: for n holds s.n>=0 & s1.n = 2 to_power n * s.(2 to_power n); A3: now let n; A4: 2 to_power n > 0 by POWER:39; s.(2 to_power n) >= 0 by A2; then 2 to_power n * s.(2 to_power n) >= 0 by A4,REAL_2:121; hence s1.n >= 0 by A2; end; thus s is summable implies s1 is summable proof assume A5: s is summable; defpred Y[Nat] means Partial_Sums(s1).$1 <= 2 * Partial_Sums(s).(2 to_power $1); A6: Y[0] proof A7: 2 to_power 0 = 0+1 by POWER:29; A8: Partial_Sums(s1).0 = s1.0 by Def1 .= 1 * s.1 by A2,A7 .= s.1; A9: 2 * Partial_Sums(s).(2 to_power 0) = 2 * (Partial_Sums(s).0 + s.1) by A7,Def1 .= 2*(s.0 + s.1) by Def1 .= 2*s.0 + 2*s.1 by XCMPLX_1:8; s.1>=0 by A2; then s.1 + s.1 >= s.1 + 0 by REAL_1:55; then A10: 2 * s.1 >= s.1 by XCMPLX_1:11; s.0>=0 by A2; then s.0 + s.0 >= 0 + 0 by REAL_1:55; then 2 * s.0 >= 0 by XCMPLX_1:11; then 2*s.0 + 2*s.1 >= s.1 + 0 by A10,REAL_1:55; hence thesis by A8,A9; end; A11: for n st Y[n] holds Y[n+1] proof let n; assume Partial_Sums(s1).n <= 2 * Partial_Sums(s).(2 to_power n); then Partial_Sums(s1).n + s1.(n+1) <= 2 * Partial_Sums(s).(2 to_power n) + s1.(n+1) by AXIOMS:24; then A12: Partial_Sums(s1).(n+1) <= 2 * Partial_Sums(s).(2 to_power n) + s1.(n+1) by Def1; deffunc U(Nat) = Partial_Sums(s).(2 to_power n + $1) - Partial_Sums(s).(2 to_power n); consider a being Real_Sequence such that A13: for m holds a.m = U(m) from ExRealSeq; defpred X[Nat] means $1 > 2 to_power n or $1 * s.(2 to_power (n+1)) <= a.$1; A14: X[0] proof a.0 = Partial_Sums(s).(2 to_power n + 0) - Partial_Sums(s).(2 to_power n) by A13 .= 0 by XCMPLX_1:14; hence thesis; end; A15: for m st X[m] holds X[m+1] proof let m; assume A16: m > 2 to_power n or m * s.(2 to_power (n+1)) <= a.m; now per cases by A16; suppose m > 2 to_power n; hence thesis by NAT_1:38; suppose A17: m * s.(2 to_power (n+1)) <= a.m; now per cases; suppose m<2 to_power n; then m+1<=2 to_power n by NAT_1:38; then 2 to_power n + (m+1) <= 2 to_power n + 2 to_power n by REAL_1:55; then 2 to_power n + m + 1 <= 2 to_power n + 2 to_power n by XCMPLX_1:1; then 2 to_power n + m + 1 <= 2 * 2 to_power n by XCMPLX_1:11; then 2 to_power n + m + 1 <= 2 to_power 1 * 2 to_power n by POWER:30; then 2 to_power n + m + 1 <= 2 to_power (n+1) by POWER:32; then s.(2 to_power n + m + 1) >= s.(2 to_power (n+1)) by A1,SEQM_3:14; then m * s.(2 to_power (n+1)) + 1 * s.(2 to_power (n+1)) <= a.m + s.(2 to_power n + m + 1) by A17,REAL_1:55; then (m + 1) * s.(2 to_power (n+1)) <= a.m + s.(2 to_power n + m + 1) by XCMPLX_1:8; then (m + 1) * s.(2 to_power (n+1)) <= Partial_Sums(s).(2 to_power n + m) - Partial_Sums(s).(2 to_power n) + s.(2 to_power n + m + 1) by A13; then (m + 1) * s.(2 to_power (n+1)) <= Partial_Sums(s).(2 to_power n + m) + s.(2 to_power n + m + 1) - Partial_Sums(s).(2 to_power n) by XCMPLX_1:29; then (m + 1) * s.(2 to_power (n+1)) <= Partial_Sums(s).(2 to_power n + m + 1) - Partial_Sums(s).(2 to_power n) by Def1; then (m + 1) * s.(2 to_power (n+1)) <= Partial_Sums(s).(2 to_power n + (m + 1)) - Partial_Sums(s).(2 to_power n) by XCMPLX_1:1; hence thesis by A13; suppose m>=2 to_power n; hence thesis by NAT_1:38; end; hence thesis; end; hence thesis; end; for m holds X[m] from Ind(A14,A15); then 2 to_power n * s.(2 to_power (n+1)) <= a.(2 to_power n); then 2*(2 to_power n * s.(2 to_power (n+1))) <= 2*a.(2 to_power n) by AXIOMS:25; then 2*2 to_power n * s.(2 to_power (n+1)) <= 2*a.(2 to_power n) by XCMPLX_1:4; then 2 to_power 1 * 2 to_power n * s.(2 to_power (n+1)) <= 2*a.(2 to_power n) by POWER:30; then 2 to_power (n+1) * s.(2 to_power (n+1)) <= 2*a.(2 to_power n) by POWER: 32 ; then s1.(n+1) <= 2*a.(2 to_power n) by A2; then A18: s1.(n+1) <= 2*(Partial_Sums(s).(2 to_power n + 2 to_power n) - Partial_Sums(s).(2 to_power n)) by A13; 2 to_power n + 2 to_power n = 2 * 2 to_power n by XCMPLX_1:11 .= 2 to_power 1 * 2 to_power n by POWER:30 .= 2 to_power (n+1) by POWER:32; then s1.(n+1) <= 2 * Partial_Sums(s).(2 to_power (n+1)) - 2 * Partial_Sums(s).(2 to_power n) by A18,XCMPLX_1:40; then 2 * Partial_Sums(s).(2 to_power n) + s1.(n+1) <= 2 * Partial_Sums(s).(2 to_power n) + (2 * Partial_Sums(s).(2 to_power (n+1)) - 2 * Partial_Sums(s).(2 to_power n)) by REAL_1:55; then 2 * Partial_Sums(s).(2 to_power n) + s1.(n+1) <= 2 * Partial_Sums(s).(2 to_power (n+1)) by XCMPLX_1:27; hence Partial_Sums(s1).(n+1) <= 2 * Partial_Sums(s).(2 to_power (n+1)) by A12,AXIOMS:22; end; A19: for n holds Y[n] from Ind(A6,A11); Partial_Sums(s) is bounded_above by A2,A5,Th20; then consider r be real number such that A20: for n holds Partial_Sums(s).n<r by SEQ_2:def 3; now let n; Partial_Sums(s).(2 to_power n)<r by A20; then A21: 2 * Partial_Sums(s).(2 to_power n) < 2*r by REAL_1:70; Partial_Sums(s1).n <= 2 * Partial_Sums(s).(2 to_power n) by A19; hence Partial_Sums(s1).n < 2*r by A21,AXIOMS:22; end; then Partial_Sums(s1) is bounded_above by SEQ_2:def 3; hence thesis by A3,Th20; end; assume A22: s1 is summable; defpred Y[Nat] means Partial_Sums(s).(2 to_power ($1+1)) <= Partial_Sums(s1).$1 + s.0 + s.2; A23: Y[0] proof A24: Partial_Sums(s).(2 to_power (0+1)) = Partial_Sums(s).(1+1) by POWER:30 .= Partial_Sums(s).(0+1)+s.2 by Def1 .= Partial_Sums(s).0+s.1+s.2 by Def1 .= s.0+s.1+s.2 by Def1; A25: 2 to_power 0 = 1 by POWER:29; Partial_Sums(s1).0 + s.0 + s.2 = s1.0 + s.0 + s.2 by Def1 .= 2 to_power 0 * s.(2 to_power 0) + s.0 + s.2 by A2 .= s.0 + s.1 + s.2 by A25; hence thesis by A24; end; A26: for n st Y[n] holds Y[n+1] proof let n; assume Partial_Sums(s).(2 to_power (n+1)) <= Partial_Sums(s1).n + s.0+s.2; then Partial_Sums(s).(2 to_power (n+1)) + s1.(n+1) <= s1.(n+1) + (Partial_Sums(s1).n + s.0+s.2) by REAL_1:55; then Partial_Sums(s).(2 to_power (n+1)) + s1.(n+1) <= s1.(n+1) + (Partial_Sums(s1).n + s.0)+s.2 by XCMPLX_1:1; then Partial_Sums(s).(2 to_power (n+1)) + s1.(n+1) <= Partial_Sums(s1).n + s1.(n+1) + s.0+s.2 by XCMPLX_1:1; then A27: Partial_Sums(s).(2 to_power (n+1)) + s1.(n+1) <= Partial_Sums(s1).(n+1) + s.0+s.2 by Def1; defpred X[Nat] means Partial_Sums(s).(2 to_power (n+1) + $1) - Partial_Sums(s).(2 to_power (n+1)) <= $1 * s.(2 to_power (n+1)); A28: X[0] by XCMPLX_1:14; A29: for m st X[m] holds X[m+1] proof let m; assume A30: Partial_Sums(s).(2 to_power (n+1) + m) - Partial_Sums(s).(2 to_power (n+1)) <= m * s.(2 to_power (n+1)); 2 to_power (n+1) + (m+1) >= 2 to_power (n+1) by NAT_1:29; then s.(2 to_power (n+1) + (m+1)) <= s.(2 to_power (n+1)) by A1,SEQM_3:14; then s.(2 to_power (n+1) + m+1) <= s.(2 to_power (n+1)) by XCMPLX_1:1; then Partial_Sums(s).(2 to_power (n+1) + m) - Partial_Sums(s).(2 to_power (n+1)) + s.(2 to_power (n+1) + m+1) <= m * s.(2 to_power (n+1)) + s.(2 to_power (n+1)) by A30,REAL_1:55; then Partial_Sums(s).(2 to_power (n+1) + m) + s.(2 to_power (n+1) + m+1) - Partial_Sums(s).(2 to_power (n+1)) <= m * s.(2 to_power (n+1)) + s.(2 to_power (n+1)) by XCMPLX_1:29; then Partial_Sums(s).(2 to_power (n+1) + m + 1) - Partial_Sums(s).(2 to_power (n+1)) <= m * s.(2 to_power (n+1)) + s.(2 to_power (n+1)) by Def1; then Partial_Sums(s).(2 to_power (n+1) + (m + 1)) - Partial_Sums(s).(2 to_power (n+1)) <= m * s.(2 to_power (n+1)) + 1*s.(2 to_power (n+1)) by XCMPLX_1:1; hence Partial_Sums(s).(2 to_power (n+1) + (m + 1)) - Partial_Sums(s).(2 to_power (n+1)) <= (m + 1) * s.(2 to_power (n+1)) by XCMPLX_1:8; end; for m holds X[m] from Ind(A28,A29); then Partial_Sums(s).(2 to_power (n+1) + 2 to_power (n+1)) - Partial_Sums(s).(2 to_power (n+1))<= 2 to_power (n+1) * s.(2 to_power (n+1)); then A31: Partial_Sums(s).(2 to_power (n+1) + 2 to_power (n+1)) - Partial_Sums(s).(2 to_power (n+1)) <= s1.(n+1) by A2; 2 to_power (n+1) + 2 to_power (n+1) = 2 * 2 to_power (n+1) by XCMPLX_1:11 .= 2 to_power 1 * 2 to_power (n+1) by POWER:30 .= 2 to_power (n+1+1) by POWER:32; then Partial_Sums(s).(2 to_power (n+1+1)) - Partial_Sums(s).(2 to_power (n+1)) + Partial_Sums(s).(2 to_power (n+1)) <= Partial_Sums(s).(2 to_power (n+1)) + s1.(n+1) by A31,REAL_1:55; then Partial_Sums(s).(2 to_power (n+1+1)) <= Partial_Sums(s).(2 to_power (n+1)) + s1.(n+1) by XCMPLX_1:27; hence Partial_Sums(s).(2 to_power (n+1+1)) <= Partial_Sums(s1).(n+1) + s.0+s.2 by A27,AXIOMS:22; end; A32: for n holds Y[n] from Ind(A23,A26); A33: Partial_Sums(s) is non-decreasing by A2,Th19; Partial_Sums(s1) is bounded_above by A3,A22,Th20; then consider r be real number such that A34: for n holds Partial_Sums(s1).n<r by SEQ_2:def 3; now let n; A35: (1+1) to_power n >= 1 + n*1 by POWER:56; 1+n>=n by NAT_1:29; then 2 to_power n >= n by A35,AXIOMS:22; then 2 to_power n + 2 to_power n>= n+n by REAL_1:55; then 2 * 2 to_power n >= n+n by XCMPLX_1:11; then 2 to_power 1 * 2 to_power n >= n+n by POWER:30; then A36: 2 to_power (n+1) >= n+n by POWER:32; n+n>=n by NAT_1:29; then 2 to_power (n+1) >= n by A36,AXIOMS:22; then A37: Partial_Sums(s).(2 to_power (n+1)) >= Partial_Sums(s).n by A33,SEQM_3:12; Partial_Sums(s).(2 to_power (n+1)) <= Partial_Sums(s1).n+s.0+s.2 by A32; then A38: Partial_Sums(s).n <= Partial_Sums(s1).n+s.0+s.2 by A37,AXIOMS:22; Partial_Sums(s1).n < r by A34; then Partial_Sums(s1).n + (s.0 + s.2) < r + (s.0+s.2) by REAL_1:53; then Partial_Sums(s1).n + s.0 + s.2 < r + (s.0+s.2) by XCMPLX_1:1; hence Partial_Sums(s).n < r + (s.0+s.2) by A38,AXIOMS:22; end; then Partial_Sums(s) is bounded_above by SEQ_2:def 3; hence thesis by A2,Th20; end; theorem p>1 & (for n st n>=1 holds s.n = 1/n to_power p) implies s is summable proof assume that A1: p>1 and A2: for n st n>=1 holds s.n = 1/n to_power p; A3: p>=0 by A1,AXIOMS:22; defpred X[Nat,real number] means ($1=0 & $2=1) or ($1>=1 & $2=1/$1 to_power p); A4: for n ex r st X[n,r] proof let n; n=0 or n>=1 proof assume A5: n<>0 & n<1; then n>0 by NAT_1:19; then n>=0+1 by NAT_1:38; hence contradiction by A5; end; hence thesis; end; consider s1 such that A6: for n holds X[n,s1.n] from RealSeqChoice(A4); A7: s1.0 = 1 by A6; A8: now let n; assume n>=0+1; then n<>0; hence s1.n=1/n to_power p by A6; end; now let n; now per cases by NAT_1:22; suppose A9: n=0; then A10: s1.(n+1) = 1/(n+1) to_power p by A8; (n+1) #R p >= 1 by A3,A9,PREPOWER:99; then (n+1) to_power p >= 1 by A9,POWER:def 2; hence s1.(n+1)<=s1.n by A7,A9,A10,REAL_2:164; suppose ex m st n=m+1; then consider m such that A11: n = m+1; A12: m>=0 by NAT_1:18; then A13: n>=0+1 by A11,REAL_1:55; A14: n+1>=0+1 by NAT_1:37; A15: n>0 by A11,A12,NAT_1:38; then A16: n to_power p > 0 by POWER:39; A17: n+1>0 by A14,NAT_1:38; then A18: n/(n+1)>0 by A15,SEQ_2:6; A19: s1.(n+1)/s1.n = (1/(n+1) to_power p)/s1.n by A8,A14 .= (1/(n+1) to_power p)/(1/n to_power p) by A8,A13 .= (1/(n+1) to_power p) * n to_power p by XCMPLX_1:101 .= n to_power p / (n+1) to_power p by XCMPLX_1:100 .= (n/(n+1)) to_power p by A15,A17,POWER:36 .= (n/(n+1)) #R p by A18,POWER:def 2; 1/n to_power p > 0 by A16,REAL_2:127; then A20: s1.n>0 by A8,A13; n<=n+1 by NAT_1:29; then n/(n+1)<=1 by A15,REAL_2:143; then s1.(n+1)/s1.n <= (n/(n+1)) #R 0 by A3,A18,A19,PREPOWER:98; then s1.(n+1)/s1.n <= 1 by A18,PREPOWER:85; hence s1.(n+1)<=s1.n by A20,REAL_2:118; end; hence s1.(n+1)<=s1.n; end; then A21: s1 is non-increasing by SEQM_3:def 14; deffunc V(Nat) = 2 to_power $1 * s1.(2 to_power $1); consider s2 such that A22: for n holds s2.n = V(n) from ExRealSeq; A23: now let n; now per cases by NAT_1:22; suppose n=0; hence s1.n >= 0 by A7; suppose ex m st n=m+1; then consider m such that A24: n = m+1; A25: m>=0 by NAT_1:18; then A26: n>=0+1 by A24,REAL_1:55; n>0 by A24,A25,NAT_1:38; then n to_power p > 0 by POWER:39; then 1/n to_power p >= 0 by REAL_2:127; hence s1.n>=0 by A8,A26; end; hence s1.n>=0 & s2.n = 2 to_power n * s1.(2 to_power n) by A22; end; deffunc U(Nat) = $1-root (s2.$1); consider s3 such that A27: for n holds s3.n = U(n) from ExRealSeq; A28: now let n; 2 to_power n > 0 by POWER:39; then A29: 2 to_power n >= 0+1 by NAT_1:38; thus A30: s2.n = 2 to_power n * s1.(2 to_power n) by A22 .= 2 to_power n * (1/(2 to_power n) to_power p) by A8,A29 .= 2 to_power n * (1/2 to_power (n*p)) by POWER:38 .= 2 to_power n * 2 to_power (-n*p) by POWER:33 .= 2 to_power (n+-n*p) by POWER:32 .= 2 to_power (n*1-n*p) by XCMPLX_0:def 8 .= 2 to_power ((1-p)*n) by XCMPLX_1:40; hence s2.n>=0 by POWER:39; s2.n = 2 to_power (1-p) to_power n by A30,POWER:38; hence s3.n = n-root (2 to_power (1-p) to_power n) by A27; end; A31: now let n; n>=0 by NAT_1:18; then A32: n+1>=0+1 by AXIOMS:24; A33: 2 to_power (1-p) <> 0 & 2 to_power (1-p) >= 0 by POWER:39; thus (s3^\1).n = s3.(n+1) by SEQM_3:def 9 .= (n+1)-root (2 to_power (1-p) to_power (n+1)) by A28 .= (n+1)-root (2 to_power (1-p) |^ (n+1)) by A33,POWER:46 .= 2 to_power (1-p) by A32,A33,POWER:5; end; then A34: s3^\1 is constant by SEQM_3:def 6; then A35: s3^\1 is convergent by SEQ_4:39; then A36: s3 is convergent by SEQ_4:35; lim (s3^\1) = (s3^\1).0 by A34,SEQ_4:41 .= 2 to_power (1-p) by A31; then A37: lim s3 = 2 to_power (1-p) by A35,SEQ_4:36; 1-p<0 & 2>1 by A1,REAL_2:105; then lim s3 < 1 by A37,POWER:41; then s2 is summable by A27,A28,A36,Th32; then s1 is summable by A21,A23,Th35; then A38: s1^\1 is summable by Th15; A39: now let n; n>=0 by NAT_1:18; then A40: n+1>=0+1 by AXIOMS:24; A41: (s^\1).n = s.(n+1) by SEQM_3:def 9 .= 1/(n+1) to_power p by A2,A40 .= s1.(n+1) by A8,A40 .= (s1^\1).n by SEQM_3:def 9; s1.(n+1)>=0 by A23; hence (s^\1).n>=0 by A41,SEQM_3:def 9; end; now let n; assume n>=0; then A42: n+1>=0+1 by AXIOMS:24; (s^\1).n = s.(n+1) by SEQM_3:def 9 .= 1/(n+1) to_power p by A2,A42 .= s1.(n+1) by A8,A42 .= (s1^\1).n by SEQM_3:def 9; hence (s1^\1).n>=(s^\1).n; end; then s^\1 is summable by A38,A39,Th22; hence thesis by Th16; end; theorem p<=1 & (for n st n>=1 holds s.n=1/n to_power p) implies s is not summable proof assume that A1: p<=1 and A2: for n st n>=1 holds s.n = 1/n to_power p; now per cases; suppose p<0; then A3: -p>=0 by REAL_1:66; now assume s is convergent & lim s=0; then consider m such that A4: for n st n>= m holds abs(s.n-0)<1 by SEQ_2:def 7; consider k such that A5: k>m by SEQ_4:10; now let n such that A6: n>=k; m>= 0 by NAT_1:18; then k>0 by A5,AXIOMS:22; then A7: n>0 by A6,AXIOMS:22; then A8: n>=0+1 by NAT_1:38; n>=m by A5,A6,AXIOMS:22; then abs(s.n-0)<1 by A4; then A9: abs(1/n to_power p)<1 by A2,A8; A10: n to_power (-p) > 0 by A7,POWER:39; abs(n to_power (-p))<1 by A7,A9,POWER:33; then A11: n to_power (-p)<1 by A10,ABSVALUE:def 1; n #R (-p) >= 1 by A3,A8,PREPOWER:99; hence contradiction by A7,A11,POWER:def 2; end; hence contradiction; end; hence s is not summable by Th7; suppose A12: p>=0; defpred X[Nat,real number] means ($1=0 & $2=1) or ($1>=1 & $2=1/$1 to_power p); A13: for n ex r st X[n,r] proof let n; n=0 or n>=1 proof assume A14: n<>0 & n<1; then n>0 by NAT_1:19; then n>=0+1 by NAT_1:38; hence contradiction by A14; end; hence thesis; end; consider s1 such that A15: for n holds X[n,s1.n] from RealSeqChoice(A13); A16: s1.0 = 1 by A15; A17: now let n; assume n>=0+1; then n<>0; hence s1.n=1/n to_power p by A15; end; now let n; now per cases by NAT_1:22; suppose A18: n=0; then A19: s1.(n+1) = 1/(n+1) to_power p by A17; (n+1) #R p >= 1 by A12,A18,PREPOWER:99; then (n+1) to_power p >= 1 by A18,POWER:def 2; hence s1.(n+1)<=s1.n by A16,A18,A19,REAL_2:164; suppose ex m st n=m+1; then consider m such that A20: n = m+1; A21: m>=0 by NAT_1:18; then A22: n>=0+1 by A20,REAL_1:55; A23: n+1>=0+1 by NAT_1:37; A24: n>0 by A20,A21,NAT_1:38; then A25: n to_power p > 0 by POWER:39; A26: n+1>0 by A23,NAT_1:38; then A27: n/(n+1)>0 by A24,SEQ_2:6; A28: s1.(n+1)/s1.n = (1/(n+1) to_power p)/s1.n by A17,A23 .= (1/(n+1) to_power p)/(1/n to_power p) by A17,A22 .= (1/(n+1) to_power p) * n to_power p by XCMPLX_1:101 .= n to_power p / (n+1) to_power p by XCMPLX_1:100 .= (n/(n+1)) to_power p by A24,A26,POWER:36 .= (n/(n+1)) #R p by A27,POWER:def 2; 1/n to_power p > 0 by A25,REAL_2:127; then A29: s1.n>0 by A17,A22; n<=n+1 by NAT_1:29; then n/(n+1)<=1 by A24,REAL_2:143; then s1.(n+1)/s1.n <= (n/(n+1)) #R 0 by A12,A27,A28,PREPOWER:98; then s1.(n+1)/s1.n <= 1 by A27,PREPOWER:85; hence s1.(n+1)<=s1.n by A29,REAL_2:118; end; hence s1.(n+1)<=s1.n; end; then A30: s1 is non-increasing by SEQM_3:def 14; deffunc U(Nat) = 2 to_power $1 * s1.(2 to_power $1); consider s2 such that A31: for n holds s2.n = U(n) from ExRealSeq; A32: now let n; now per cases by NAT_1:22; suppose n=0; hence s1.n >= 0 by A16; suppose ex m st n=m+1; then consider m such that A33: n = m+1; A34: m>=0 by NAT_1:18; then A35: n>=0+1 by A33,REAL_1:55; n>0 by A33,A34,NAT_1:38; then n to_power p > 0 by POWER:39; then 1/n to_power p >= 0 by REAL_2:127; hence s1.n>=0 by A17,A35; end; hence s1.n>=0 & s2.n = 2 to_power n * s1.(2 to_power n) by A31; end; now assume s2 is convergent & lim s2=0; then consider m such that A36: for n st n>=m holds abs(s2.n-0)<1/2 by SEQ_2: def 7; now let n; assume n>=m; then abs(s2.n-0)<1/2 by A36; then A37: abs(2 to_power n * s1.(2 to_power n))<1/2 by A31; 2 to_power n = 2 |^ n by POWER:46; then 2 to_power n >= 1 by PREPOWER:19; then abs(2 to_power n * (1/(2 to_power n) to_power p))<1/2 by A17,A37; then abs(2 to_power n * (1/2 to_power (n*p)))<1/2 by POWER:38; then abs(2 to_power n * 2 to_power (-n*p))<1/2 by POWER:33; then abs(2 to_power (n+-n*p))<1/2 by POWER:32; then A38: abs(2 to_power (n-n*p))<1/2 by XCMPLX_0:def 8; 2 to_power (n-n*p)>0 by POWER:39; then 2 to_power (n-n*p)<1/2 by A38,ABSVALUE:def 1; then 2 to_power (n-n*p)*2<1/2*2 by REAL_1:70; then 2 to_power (n-n*p)*2 to_power 1<1 by POWER:30; then A39: 2 to_power (n-n*p+1)<1 by POWER:32; A40: 1-p>=0 by A1,SQUARE_1:12; n>=0 by NAT_1:18; then n*(1-p)>=0 by A40,REAL_2:121; then n*1-n*p>=0 by XCMPLX_1:40; then n - n*p + 1 >= 0+0 by REAL_1:55; then 2 #R (n-n*p+1) >= 1 by PREPOWER:99; hence contradiction by A39,POWER:def 2; end; hence contradiction; end; then not s2 is summable by Th7; then A41: not s1 is summable by A30,A32,Th35; now let n; assume A42: n>=1; then s.n = 1/n to_power p by A2 .= s1.n by A17,A42; hence s.n>=s1.n; end; hence s is not summable by A32,A41,Th22; end; hence thesis; end; definition let s; canceled; attr s is absolutely_summable means :Def5: abs(s) is summable; end; canceled; theorem Th39: for n,m st n<=m holds abs(Partial_Sums(s).m - Partial_Sums(s).n) <= abs(Partial_Sums(abs(s)).m - Partial_Sums(abs(s)).n) proof let n,m such that A1: n<=m; set s1=Partial_Sums(s); set s2=Partial_Sums(abs(s)); defpred X[Nat] means abs(s1.(n+$1) - s1.n) <= abs(s2.(n+$1) - s2.n); abs(s1.(n+0) - s1.n) = abs(0) by XCMPLX_1:14 .= 0 by ABSVALUE:7; then A2: X[0] by ABSVALUE:5; now let k; abs(s.k)>=0 by ABSVALUE:5; hence abs(s).k >= 0 by SEQ_1:16; end; then A3: s2 is non-decreasing by Th19; A4: for k st X[k] holds X[k+1] proof let k; assume A5: abs(s1.(n+k) - s1.n) <= abs(s2.(n+k) - s2.n); abs(s1.(n+(k+1)) - s1.n) = abs(s1.(n+k+1) - s1.n) by XCMPLX_1:1 .= abs(s.(n+k+1) + s1.(n+k) - s1.n) by Def1 .= abs(s.(n+k+1) + (s1.(n+k) - s1.n)) by XCMPLX_1:29; then A6: abs(s1.(n+(k+1))-s1.n) <= abs(s.(n+k+1))+abs(s1.(n+k) - s1.n) by ABSVALUE:13; abs(s.(n+k+1)) + abs(s1.(n+k) - s1.n) <= abs(s.(n+k+1)) + abs(s2.(n+k) - s2.n) by A5,REAL_1:55; then A7: abs(s1.(n+(k+1))-s1.n) <= abs(s.(n+k+1))+abs(s2.(n+k)-s2.n) by A6,AXIOMS:22; s2.(n+k)>=s2.n by A3,SEQM_3:11; then A8: s2.(n+k) - s2.n >= 0 by SQUARE_1:12; then A9: abs(s1.(n+(k+1))-s1.n) <= abs(s.(n+k+1))+(s2.(n+k)-s2.n) by A7, ABSVALUE:def 1; abs(s.(n+k+1)) >= 0 by ABSVALUE:5; then A10: abs(s.(n+k+1))+(s2.(n+k)-s2.n) >= 0+0 by A8,REAL_1:55; abs(s2.(n+(k+1)) - s2.n) = abs(s2.(n+k+1) - s2.n) by XCMPLX_1:1 .= abs(s2.(n+k) + (abs(s)).(n+k+1) - s2.n) by Def1 .= abs(abs(s.(n+k+1)) + s2.(n+k) - s2.n) by SEQ_1:16 .= abs(abs(s.(n+k+1)) + (s2.(n+k) - s2.n)) by XCMPLX_1: 29; hence abs(s1.(n+(k+1))-s1.n) <= abs(s2.(n+(k+1))-s2.n) by A9,A10,ABSVALUE:def 1; end; A11: for k holds X[k] from Ind(A2,A4); reconsider u=n, v=m as Integer; reconsider k = v-u as Nat by A1,INT_1:18; n+k = m by XCMPLX_1:27; hence thesis by A11; end; theorem s is absolutely_summable implies s is summable proof assume s is absolutely_summable; then A1: abs(s) is summable by Def5; now let r; assume 0<r; then consider n such that A2: for m st n<=m holds abs(Partial_Sums(abs(s)).m - Partial_Sums(abs(s)).n)<r by A1,Th25; take n; let m; assume A3: n<=m; then A4: abs(Partial_Sums(abs(s)).m - Partial_Sums(abs(s)).n)<r by A2; abs(Partial_Sums(s).m - Partial_Sums(s).n) <= abs(Partial_Sums(abs(s)).m - Partial_Sums(abs(s)).n) by A3,Th39; hence abs(Partial_Sums(s).m - Partial_Sums(s).n)<r by A4,AXIOMS:22; end; hence thesis by Th25; end; theorem (for n holds 0<=s.n) & s is summable implies s is absolutely_summable proof assume that A1: for n holds 0<=s.n and A2: s is summable; A3: Partial_Sums(s) is convergent by A2,Def2; now let n; 0<=s.n by A1; hence s.n = abs(s.n) by ABSVALUE:def 1; end; then Partial_Sums(abs(s)) is convergent by A3,SEQ_1:16; then abs(s) is summable by Def2; hence thesis by Def5; end; theorem (for n holds s.n<>0 & s1.n=abs(s).(n+1)/abs(s).n) & s1 is convergent & lim s1 < 1 implies s is absolutely_summable proof assume that A1: for n holds s.n<>0 & s1.n=abs(s).(n+1)/abs(s).n and A2: s1 is convergent & lim s1 < 1; now let n; A3: abs(s).n = abs(s.n) by SEQ_1:16; s.n <> 0 by A1; hence abs(s).n > 0 by A3,ABSVALUE:6; thus s1.n=abs(s).(n+1)/abs(s).n by A1; end; then abs(s) is summable by A2,Th30; hence thesis by Def5; end; theorem Th43: r>0 & (ex m st for n st n>=m holds abs(s.n)>=r) implies s is not convergent or lim s <> 0 proof assume A1: r>0; given m such that A2: for n st n>=m holds abs(s.n)>=r; now per cases; suppose s is not convergent; hence thesis; suppose A3: s is convergent; now assume lim s=0; then consider k such that A4: for n st n>=k holds abs(s.n-0)<r by A1,A3,SEQ_2:def 7; now let n such that A5: n>=m+k; m+k>=m by NAT_1:29; then A6: n>=m by A5,AXIOMS:22; m+k>=k by NAT_1:29; then n>=k by A5,AXIOMS:22; then abs(s.n-0)<r by A4; hence contradiction by A2,A6; end; hence contradiction; end; hence thesis; end; hence thesis; end; theorem (for n holds s.n<>0) & (ex m st for n st n>=m holds abs(s).(n+1)/abs(s).n >= 1) implies s is not summable proof assume A1: for n holds s.n<>0; given m such that A2: for n st n>=m holds abs(s).(n+1)/abs(s).n >= 1; s.m <> 0 by A1; then A3: abs(s.m) > 0 by ABSVALUE:6; now let n such that A4: n>=m; defpred X[Nat] means abs(s.(m+$1))>=abs(s.m); A5: X[0]; A6: for k st X[k] holds X[k+1] proof let k such that A7: abs(s.(m+k))>=abs(s.m); m+k>=m by NAT_1:29; then abs(s).(m+k+1)/abs(s).(m+k) >= 1 by A2; then abs(s.(m+k+1))/abs(s).(m+k) >= 1 by SEQ_1:16; then A8: abs(s.(m+k+1))/abs(s.(m+k)) >= 1 by SEQ_1:16; s.(m+k) <> 0 by A1; then abs(s.(m+k)) > 0 by ABSVALUE:6; then abs(s.(m+k+1))>=abs(s.(m+k)) by A8,REAL_2:118; then abs(s.(m+k+1))>=abs(s.m) by A7,AXIOMS:22; hence thesis by XCMPLX_1:1; end; A9: for k holds X[k] from Ind(A5,A6); ex k st n=m+k by A4,NAT_1:28; hence abs(s.n)>=abs(s.m) by A9; end; then s is not convergent or lim s <> 0 by A3,Th43; hence thesis by Th7; end; theorem (for n holds s1.n = n-root (abs(s).n)) & s1 is convergent & lim s1 < 1 implies s is absolutely_summable proof assume that A1: for n holds s1.n = n-root (abs(s).n) and A2: s1 is convergent & lim s1 < 1; now let n; abs(s).n = abs(s.n) by SEQ_1:16; hence abs(s).n >= 0 by ABSVALUE:5; thus s1.n = n-root (abs(s).n) by A1; end; then abs(s) is summable by A2,Th32; hence thesis by Def5; end; theorem (for n holds s1.n = n-root(abs(s).n)) & (ex m st for n st m<=n holds s1.n>=1) implies s is not summable proof assume A1: for n holds s1.n = n-root (abs(s).n); given m such that A2: for n st m<=n holds s1.n>=1; now let n such that A3: n>=m+1; m+1>=m by NAT_1:29; then A4: n>=m by A3,AXIOMS:22; A5: abs(s.n) >= 0 by ABSVALUE:5; s1.n = n-root (abs(s).n) by A1 .= n-root abs(s.n) by SEQ_1:16; then n-root abs(s.n) >= 1 by A2,A4; then A6: n-root abs(s.n) |^ n >= 1 by PREPOWER:19; m+1>=1 by NAT_1:29; then n>=1 by A3,AXIOMS:22; hence abs(s.n) >= 1 by A5,A6,POWER:5; end; then s is not convergent or lim s<>0 by Th43; hence thesis by Th7; end; theorem (for n holds s1.n = n-root (abs(s).n)) & s1 is convergent & lim s1 > 1 implies s is not summable proof assume that A1: for n holds s1.n = n-root (abs(s).n) and A2: s1 is convergent & lim s1 > 1; lim s1 - 1 > 0 by A2,SQUARE_1:11; then consider m such that A3: for n st n>=m holds abs(s1.n-lim s1)<lim s1-1 by A2,SEQ_2:def 7; now let n such that A4: n>=m+1; m+1>=m by NAT_1:29; then A5: n>=m by A4,AXIOMS:22; A6: abs(s.n) >= 0 by ABSVALUE:5; s1.n = n-root (abs(s).n) by A1 .= n-root abs(s.n) by SEQ_1:16; then abs(n-root abs(s.n) - lim s1) < lim s1 - 1 by A3,A5; then -(lim s1 - 1) < n-root abs(s.n) - lim s1 by SEQ_2:9; then 1-lim s1 < n-root abs(s.n) - lim s1 by XCMPLX_1:143; then 1 - lim s1 + lim s1 < n-root abs(s.n) - lim s1 + lim s1 by REAL_1:53; then 1 < n-root abs(s.n) - lim s1 + lim s1 by XCMPLX_1:27; then n-root abs(s.n) >= 1 by XCMPLX_1:27; then A7: n-root abs(s.n) |^ n >= 1 by PREPOWER:19; m+1>=1 by NAT_1:29; then n>=1 by A4,AXIOMS:22; hence abs(s.n) >= 1 by A6,A7,POWER:5; end; then s is not convergent or lim s<>0 by Th43; hence thesis by Th7; end;

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