Volume 3, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Series

**by****Konrad Raczkowski, and****Andrzej Nedzusiak**- Received October 15, 1990
- MML identifier: SERIES_1

- [ Mizar article, 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; begin reserve n,m,k for Nat; reserve a,p,r for real number; reserve s,s1,s2,s3 for Real_Sequence; theorem :: SERIES_1:1 0<a & a<1 & (for n holds s.n=a to_power (n+1)) implies s is convergent & lim s = 0; theorem :: SERIES_1:2 a <> 0 implies abs(a) to_power n = abs(a to_power n); theorem :: SERIES_1:3 abs(a)<1 & (for n holds s.n=a to_power (n+1)) implies s is convergent & lim s = 0; definition let s; func Partial_Sums(s) -> Real_Sequence means :: SERIES_1:def 1 it.0=s.0 & for n holds it.(n+1) = it.n + s.(n+1); end; definition let s; attr s is summable means :: SERIES_1:def 2 Partial_Sums(s) is convergent; func Sum(s) -> Real equals :: SERIES_1:def 3 lim Partial_Sums(s); end; canceled 3; theorem :: SERIES_1:7 s is summable implies s is convergent & lim s = 0; theorem :: SERIES_1:8 Partial_Sums(s1) + Partial_Sums(s2) = Partial_Sums(s1+s2); theorem :: SERIES_1:9 Partial_Sums(s1) - Partial_Sums(s2) = Partial_Sums(s1-s2); theorem :: SERIES_1:10 s1 is summable & s2 is summable implies s1+s2 is summable & Sum(s1+s2) = Sum(s1) + Sum(s2); theorem :: SERIES_1:11 s1 is summable & s2 is summable implies s1-s2 is summable & Sum(s1-s2) = Sum(s1) - Sum(s2); theorem :: SERIES_1:12 Partial_Sums(r(#)s) = r(#)Partial_Sums(s); theorem :: SERIES_1:13 s is summable implies r(#)s is summable & Sum(r(#)s) =r*Sum(s); theorem :: SERIES_1:14 for s,s1 st for n holds s1.n=s.0 holds Partial_Sums(s^\1) = (Partial_Sums(s)^\1) - s1; theorem :: SERIES_1:15 s is summable implies for n holds s^\n is summable; theorem :: SERIES_1:16 (ex n st s^\n is summable) implies s is summable; theorem :: SERIES_1:17 (for n holds s1.n<=s2.n) implies for n holds Partial_Sums(s1).n<=Partial_Sums(s2).n; theorem :: SERIES_1:18 s is summable implies for n holds Sum(s) = Partial_Sums(s).n + Sum(s^\(n+1)); theorem :: SERIES_1:19 (for n holds 0<=s.n) implies Partial_Sums(s) is non-decreasing; theorem :: SERIES_1:20 (for n holds 0<=s.n) implies (Partial_Sums(s) is bounded_above iff s is summable); theorem :: SERIES_1:21 s is summable & (for n holds 0<=s.n) implies 0<=Sum(s); theorem :: SERIES_1:22 (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; canceled; theorem :: SERIES_1:24 (for n holds 0<=s1.n & s1.n<=s2.n) & s2 is summable implies s1 is summable & Sum(s1)<=Sum(s2); theorem :: SERIES_1:25 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; theorem :: SERIES_1:26 a <> 1 implies Partial_Sums(a GeoSeq).n = (1 - a to_power (n+1))/(1-a); theorem :: SERIES_1:27 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); theorem :: SERIES_1:28 abs(a)<1 implies a GeoSeq is summable & Sum(a GeoSeq) = 1/(1-a); theorem :: SERIES_1:29 abs(a) < 1 & (for n holds s.(n+1) = a * s.n) implies s is summable & Sum(s) = s.0/(1-a); theorem :: SERIES_1:30 (for n holds s.n>0 & s1.n=s.(n+1)/s.n) & s1 is convergent & lim s1 < 1 implies s is summable; theorem :: SERIES_1:31 (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; theorem :: SERIES_1:32 (for n holds s.n>=0 & s1.n = n-root (s.n)) & s1 is convergent & lim s1 < 1 implies s is summable; theorem :: SERIES_1:33 (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; theorem :: SERIES_1:34 (for n holds s.n>=0 & s1.n = n-root (s.n)) & s1 is convergent & lim s1 > 1 implies s is not summable; definition let k, n be Nat; redefine func k to_power n -> Nat; end; theorem :: SERIES_1:35 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); theorem :: SERIES_1:36 p>1 & (for n st n>=1 holds s.n = 1/n to_power p) implies s is summable; theorem :: SERIES_1:37 p<=1 & (for n st n>=1 holds s.n=1/n to_power p) implies s is not summable; definition let s; canceled; attr s is absolutely_summable means :: SERIES_1:def 5 abs(s) is summable; end; canceled; theorem :: SERIES_1:39 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); theorem :: SERIES_1:40 s is absolutely_summable implies s is summable; theorem :: SERIES_1:41 (for n holds 0<=s.n) & s is summable implies s is absolutely_summable; theorem :: SERIES_1:42 (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; theorem :: SERIES_1:43 r>0 & (ex m st for n st n>=m holds abs(s.n)>=r) implies s is not convergent or lim s <> 0; theorem :: SERIES_1:44 (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; theorem :: SERIES_1:45 (for n holds s1.n = n-root (abs(s).n)) & s1 is convergent & lim s1 < 1 implies s is absolutely_summable; theorem :: SERIES_1:46 (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; theorem :: SERIES_1:47 (for n holds s1.n = n-root (abs(s).n)) & s1 is convergent & lim s1 > 1 implies s is not summable;

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