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Theorem List for Metamath Proof Explorer - 12201-12300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfsumsub 12201* Split a finite sum over a subtraction. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  ( B  -  C )  =  ( sum_ k  e.  A  B  -  sum_ k  e.  A  C ) )
 
Theoremfsum2mul 12202* Separate the nested sum of the product  C ( j )  x.  D ( k ). (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  C  e.  CC )   &    |-  ( ( ph  /\  k  e.  B ) 
 ->  D  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_
 k  e.  B  ( C  x.  D )  =  ( sum_ j  e.  A  C  x.  sum_ k  e.  B  D ) )
 
Theoremfsumconst 12203* The sum of constant terms ( k is not free in  A). (Contributed by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ( A  e.  Fin  /\  B  e.  CC )  -> 
 sum_ k  e.  A  B  =  ( ( # `
  A )  x.  B ) )
 
Theoremfsumge0 12204* If all of the terms of a finite sum are nonnegative, so is the sum. (Contributed by NM, 26-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  0  <_  B )   =>    |-  ( ph  ->  0  <_ 
 sum_ k  e.  A  B )
 
Theoremfsumless 12205* A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by NM, 26-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  0  <_  B )   &    |-  ( ph  ->  C 
 C_  A )   =>    |-  ( ph  ->  sum_
 k  e.  C  B  <_ 
 sum_ k  e.  A  B )
 
Theoremfsumge1 12206* A sum of nonnegative numbers is greater than or equal to any one of its terms. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 4-Jun-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  0  <_  B )   &    |-  ( k  =  M  ->  B  =  C )   &    |-  ( ph  ->  M  e.  A )   =>    |-  ( ph  ->  C 
 <_  sum_ k  e.  A  B )
 
Theoremfsum00 12207* A sum of nonnegative numbers is zero iff all terms are zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  0  <_  B )   =>    |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  <->  A. k  e.  A  B  =  0 )
 )
 
Theoremfsumle 12208* If all of the terms of finite sums compare, so do the sums. (Contributed by NM, 11-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  <_  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  <_  sum_ k  e.  A  C )
 
Theoremfsumlt 12209* If every term in one finite sum is less than the corresponding term in another, then the first sum is less than the second. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  <  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  <  sum_ k  e.  A  C )
 
Theoremfsumabs 12210* Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  sum_ k  e.  A  B )  <_  sum_ k  e.  A  ( abs `  B )
 )
 
Theoremfsumtscopo 12211* Sum of a telescoping series, using half-open intervals. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( k  =  j 
 ->  A  =  B )   &    |-  ( k  =  (
 j  +  1 ) 
 ->  A  =  C )   &    |-  ( k  =  M  ->  A  =  D )   &    |-  ( k  =  N  ->  A  =  E )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  ( M..^ N ) ( B  -  C )  =  ( D  -  E ) )
 
Theoremfsumtscopo2 12212* Sum of a telescoping series. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( k  =  j 
 ->  A  =  B )   &    |-  ( k  =  (
 j  +  1 ) 
 ->  A  =  C )   &    |-  ( k  =  M  ->  A  =  D )   &    |-  ( k  =  N  ->  A  =  E )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  ( M..^ N ) ( C  -  B )  =  ( E  -  D ) )
 
Theoremfsumtscop 12213* Sum of a telescoping series. (Contributed by Scott Fenton, 24-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)
 |-  ( k  =  j 
 ->  A  =  B )   &    |-  ( k  =  (
 j  +  1 ) 
 ->  A  =  C )   &    |-  ( k  =  M  ->  A  =  D )   &    |-  ( k  =  ( N  +  1 )  ->  A  =  E )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( N  +  1 ) ) ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) ( B  -  C )  =  ( D  -  E ) )
 
Theoremfsumtscop2 12214* Sum of a telescoping series. (Contributed by Mario Carneiro, 15-Jun-2014.) (Revised by Mario Carneiro, 2-May-2016.)
 |-  ( k  =  j 
 ->  A  =  B )   &    |-  ( k  =  (
 j  +  1 ) 
 ->  A  =  C )   &    |-  ( k  =  M  ->  A  =  D )   &    |-  ( k  =  ( N  +  1 )  ->  A  =  E )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( N  +  1 ) ) ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) ( C  -  B )  =  ( E  -  D ) )
 
Theoremfsumparts 12215* Summation by parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( k  =  j 
 ->  ( A  =  B  /\  V  =  W ) )   &    |-  ( k  =  ( j  +  1 )  ->  ( A  =  C  /\  V  =  X ) )   &    |-  (
 k  =  M  ->  ( A  =  D  /\  V  =  Y )
 )   &    |-  ( k  =  N  ->  ( A  =  E  /\  V  =  Z ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( M ... N ) )  ->  V  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  ( M..^ N ) ( B  x.  ( X  -  W ) )  =  ( ( ( E  x.  Z )  -  ( D  x.  Y ) )  -  sum_
 j  e.  ( M..^ N ) ( ( C  -  B )  x.  X ) ) )
 
Theoremfsumrelem 12216* Lemma for fsumre 12217, fsumim 12218, and fsumcj 12219. (Contributed by Mario Carneiro, 25-Jul-2014.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  F : CC --> CC   &    |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( F `  ( x  +  y ) )  =  ( ( F `  x )  +  ( F `  y ) ) )   =>    |-  ( ph  ->  ( F `  sum_ k  e.  A  B )  =  sum_ k  e.  A  ( F `
  B ) )
 
Theoremfsumre 12217* The real part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  sum_
 k  e.  A  B )  =  sum_ k  e.  A  ( Re `  B ) )
 
Theoremfsumim 12218* The imaginary part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  sum_
 k  e.  A  B )  =  sum_ k  e.  A  ( Im `  B ) )
 
Theoremfsumcj 12219* The complex conjugate of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( * `  sum_
 k  e.  A  B )  =  sum_ k  e.  A  ( * `  B ) )
 
Theoremfsumrlim 12220* Limit of a finite sum of converging sequences. Note that  C
( k ) is a collection of functions with implicit parameter  k, each of which converges to  D ( k ) as  n  ~~>  +oo. (Contributed by Mario Carneiro, 22-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  k  e.  B ) )  ->  C  e.  V )   &    |-  (
 ( ph  /\  k  e.  B )  ->  ( x  e.  A  |->  C )  ~~> r  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  sum_ k  e.  B  C )  ~~> r  sum_ k  e.  B  D )
 
Theoremfsumo1 12221* The finite sum of eventually bounded functions (where the index set  B does not depend on  x) is eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 22-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  k  e.  B ) )  ->  C  e.  V )   &    |-  (
 ( ph  /\  k  e.  B )  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  sum_ k  e.  B  C )  e.  O ( 1 ) )
 
Theoremo1fsum 12222* If  A (
k ) is O(1), then  sum_ k  <_  x ,  A (
k ) is O( x). (Contributed by Mario Carneiro, 23-May-2016.)
 |-  ( ( ph  /\  k  e.  NN )  ->  A  e.  V )   &    |-  ( ph  ->  ( k  e.  NN  |->  A )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ k  e.  ( 1 ... ( |_ `  x ) ) A  /  x ) )  e.  O ( 1 ) )
 
Theoremseqabs 12223* Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by Mario Carneiro, 26-Mar-2014.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( G `  k
 )  =  ( abs `  ( F `  k
 ) ) )   =>    |-  ( ph  ->  ( abs `  (  seq  M (  +  ,  F ) `  N ) ) 
 <_  (  seq  M (  +  ,  G ) `
  N ) )
 
Theoremiserabs 12224* Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  seq 
 M (  +  ,  F )  ~~>  A )   &    |-  ( ph  ->  seq  M (  +  ,  G )  ~~>  B )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( abs `  ( F `  k
 ) ) )   =>    |-  ( ph  ->  ( abs `  A )  <_  B )
 
Theoremcvgcmp 12225* A comparison test for convergence of a real infinite series. Exercise 3 of [Gleason] p. 182. (Contributed by NM, 1-May-2005.) (Revised by Mario Carneiro, 24-Mar-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  N ) )  -> 
 0  <_  ( G `  k ) )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  N )
 )  ->  ( G `  k )  <_  ( F `  k ) )   =>    |-  ( ph  ->  seq  M (  +  ,  G )  e.  dom  ~~>  )
 
Theoremcvgcmpub 12226* An upper bound for the limit of a real infinite series. This theorem can also be used to compare two infinite series. (Contributed by Mario Carneiro, 24-Mar-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  A )   &    |-  ( ph  ->  seq 
 M (  +  ,  G )  ~~>  B )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k ) )   =>    |-  ( ph  ->  B  <_  A )
 
Theoremcvgcmpce 12227* A comparison test for convergence of a complex infinite series. (Contributed by NM, 25-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  CC )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  N )
 )  ->  ( abs `  ( G `  k
 ) )  <_  ( C  x.  ( F `  k ) ) )   =>    |-  ( ph  ->  seq  M (  +  ,  G )  e.  dom  ~~>  )
 
Theoremabscvgcvg 12228* An absolutely convergent series is convergent. (Contributed by Mario Carneiro, 28-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( abs `  ( G `  k ) ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  CC )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  seq  M (  +  ,  G )  e.  dom  ~~>  )
 
Theoremclimfsum 12229* Limit of a finite sum of converging sequences. Note that  F
( k ) is a collection of functions with implicit parameter  k, each of which converges to  B ( k ) as  n  ~~>  +oo. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Mario Carneiro, 22-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  F  ~~>  B )   &    |-  ( ph  ->  H  e.  W )   &    |-  (
 ( ph  /\  ( k  e.  A  /\  n  e.  Z ) )  ->  ( F `  n )  e.  CC )   &    |-  (
 ( ph  /\  n  e.  Z )  ->  ( H `  n )  = 
 sum_ k  e.  A  ( F `  n ) )   =>    |-  ( ph  ->  H  ~~>  sum_
 k  e.  A  B )
 
Theoremfsumiun 12230* Sum over a disjoint indexed union. (Contributed by Mario Carneiro, 1-Jul-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  Fin )   &    |-  ( ph  -> Disj  x  e.  A B )   &    |-  ( ( ph  /\  ( x  e.  A  /\  k  e.  B )
 )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  U_  x  e.  A  B C  =  sum_ x  e.  A  sum_ k  e.  B  C )
 
Theoremhashiun 12231* The cardinality of a disjoint indexed union. (Contributed by Mario Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  Fin )   &    |-  ( ph  -> Disj  x  e.  A B )   =>    |-  ( ph  ->  ( # `
  U_ x  e.  A  B )  =  sum_ x  e.  A  ( # `  B ) )
 
TheoremfsumiunOLD 12232* Sum over a disjoint indexed union. (Contributed by Mario Carneiro, 1-Jul-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  Fin )   &    |-  ( ph  ->  E* x ( x  e.  A  /\  y  e.  B ) )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  k  e.  B ) )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  U_  x  e.  A  B C  =  sum_ x  e.  A  sum_ k  e.  B  C )
 
TheoremhashiunOLD 12233* The cardinality of a disjoint indexed union. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  Fin )   &    |-  ( ph  ->  E* x ( x  e.  A  /\  y  e.  B ) )   =>    |-  ( ph  ->  ( # `  U_ x  e.  A  B )  = 
 sum_ x  e.  A  ( # `  B ) )
 
Theoremhashuni 12234* The cardinality of a disjoint union. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Fin )   &    |-  ( ph  -> Disj  x  e.  A x )   =>    |-  ( ph  ->  ( # `  U. A )  =  sum_ x  e.  A  ( # `  x ) )
 
TheoremhashuniOLD 12235* The cardinality of a disjoint union. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Fin )   &    |-  ( ph  ->  E* x ( x  e.  A  /\  y  e.  x ) )   =>    |-  ( ph  ->  ( # `  U. A )  =  sum_ x  e.  A  ( # `  x ) )
 
Theoremqshash 12236* The cardinality of a set with an equivalence relation is the sum of the cardinalities of its equivalence classes. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( ph  ->  .~  Er  A )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  ( # `  A )  = 
 sum_ x  e.  ( A /.  .~  ) ( # `  x ) )
 
Theoremackbijnn 12237* Translate the Ackermann bijection ackbij1 7818 onto the natural numbers. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  F  =  ( x  e.  ( ~P NN0  i^i 
 Fin )  |->  sum_ y  e.  x  ( 2 ^ y ) )   =>    |-  F : ( ~P NN0  i^i 
 Fin ) -1-1-onto-> NN0
 
5.8.4  The binomial theorem
 
Theorembinomlem 12238* Lemma for binom 12239 (binomial theorem). Inductive step. (Contributed by NM, 6-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ps  ->  (
 ( A  +  B ) ^ N )  = 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( A ^ ( N  -  k ) )  x.  ( B ^
 k ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( A  +  B ) ^
 ( N  +  1 ) )  =  sum_ k  e.  ( 0 ... ( N  +  1 ) ) ( ( ( N  +  1 )  _C  k )  x.  ( ( A ^ ( ( N  +  1 )  -  k ) )  x.  ( B ^ k
 ) ) ) )
 
Theorembinom 12239* The binomial theorem:  ( A  +  B
) ^ N is the sum from  k  =  0 to  N of  ( N  _C  k )  x.  ( ( A ^
k )  x.  ( B ^ ( N  -  k ) ). Theorem 15-2.8 of [Gleason] p. 296. This part of the proof sets up the induction and does the base case, with the bulk of the work (the induction step) in binomlem 12238. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  +  B ) ^ N )  =  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k )  x.  ( ( A ^
 ( N  -  k
 ) )  x.  ( B ^ k ) ) ) )
 
Theorembinom1p 12240* Special case of the binomial theorem for  ( 1  +  A
) ^ N. (Contributed by Paul Chapman, 10-May-2007.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( ( 1  +  A ) ^ N )  =  sum_ k  e.  ( 0 ...
 N ) ( ( N  _C  k )  x.  ( A ^
 k ) ) )
 
Theorembinom11 12241* Special case of the binomial theorem for  2 ^ N. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( N  e.  NN0  ->  ( 2 ^ N )  =  sum_ k  e.  ( 0 ... N ) ( N  _C  k ) )
 
Theorembinom1dif 12242* A summation for the difference between  ( ( A  + 
1 ) ^ N
) and  ( A ^ N ). (Contributed by Scott Fenton, 9-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( ( ( A  +  1 ) ^ N )  -  ( A ^ N ) )  =  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k ) ) )
 
Theorembcxmaslem1 12243 Lemma for bcxmas 12245. (Contributed by Paul Chapman, 18-May-2007.)
 |-  ( A  =  B  ->  ( ( N  +  A )  _C  A )  =  ( ( N  +  B )  _C  B ) )
 
Theorembcxmaslem2 12244 Lemma for bcxmas 12245. (Contributed by Paul Chapman, 18-May-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C )
 )  =  ( ( A  +  C )  +  B ) )
 
Theorembcxmas 12245* Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( ( N  +  1 )  +  M )  _C  M )  =  sum_ j  e.  (
 0 ... M ) ( ( N  +  j
 )  _C  j )
 )
 
5.8.5  Infinite sums (cont.)
 
Theoremisumshft 12246* Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  ( M  +  K ) )   &    |-  (
 j  =  ( K  +  k )  ->  A  =  B )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  W ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  W  A  =  sum_ k  e.  Z  B )
 
Theoremisumsplit 12247* Split off the first  N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  =  (
 sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A ) )
 
Theoremisum1p 12248* The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  =  ( ( F `  M )  +  sum_ k  e.  ( ZZ>= `  ( M  +  1 ) ) A ) )
 
Theoremisumnn0nn 12249* Sum from 0 to infinity in terms of sum from 1 to infinity. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( k  =  0 
 ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  A  e.  CC )   &    |-  ( ph  ->  seq  0 (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  NN0  A  =  ( B  +  sum_ k  e.  NN  A ) )
 
Theoremisumrpcl 12250* The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  RR+ )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  W  A  e.  RR+ )
 
Theoremisumle 12251* Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  A  <_  B )   &    |-  ( ph  ->  seq 
 M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq  M (  +  ,  G )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  <_  sum_ k  e.  Z  B )
 
Theoremisumless 12252* A finite sum of nonnegative numbers is less or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  B )   &    |-  ( ( ph  /\  k  e.  Z )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  B )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  A  B  <_  sum_ k  e.  Z  B )
 
Theoremisumsup2 12253* An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  G  =  seq  M (  +  ,  F )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  A )   &    |-  ( ( ph  /\  k  e.  Z )  ->  A  e.  RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  A )   &    |-  ( ph  ->  E. x  e.  RR  A. j  e.  Z  ( G `  j )  <_  x )   =>    |-  ( ph  ->  G  ~~>  sup ( ran 
 G ,  RR ,  <  ) )
 
Theoremisumsup 12254* An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  G  =  seq  M (  +  ,  F )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  A )   &    |-  ( ( ph  /\  k  e.  Z )  ->  A  e.  RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  A )   &    |-  ( ph  ->  E. x  e.  RR  A. j  e.  Z  ( G `  j )  <_  x )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  =  sup ( ran  G ,  RR ,  <  ) )
 
Theoremisumltss 12255* A partial sum of a series with positive terms is less than the infinite sum. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  B )   &    |-  ( ( ph  /\  k  e.  Z )  ->  B  e.  RR+ )   &    |-  ( ph  ->  seq 
 M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  A  B  <  sum_ k  e.  Z  B )
 
Theoremclimcndslem1 12256* Lemma for climcnds 12258: bound the original series by the condensed series. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  NN )  ->  0  <_  ( F `  k ) )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  n  e.  NN0 )  ->  ( G `  n )  =  ( (
 2 ^ n )  x.  ( F `  ( 2 ^ n ) ) ) )   =>    |-  ( ( ph  /\  N  e.  NN0 )  ->  (  seq  1 (  +  ,  F ) `  (
 ( 2 ^ ( N  +  1 )
 )  -  1 ) )  <_  (  seq  0 (  +  ,  G ) `  N ) )
 
Theoremclimcndslem2 12257* Lemma for climcnds 12258: bound the condensed series by the original series. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  NN )  ->  0  <_  ( F `  k ) )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  n  e.  NN0 )  ->  ( G `  n )  =  ( (
 2 ^ n )  x.  ( F `  ( 2 ^ n ) ) ) )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  (  seq  1 (  +  ,  G ) `  N )  <_  ( 2  x.  (  seq  1 (  +  ,  F ) `
  ( 2 ^ N ) ) ) )
 
Theoremclimcnds 12258* The Cauchy condensation test. If  a ( k ) is a decreasing sequence of nonnegative terms, then  sum_ k  e.  NN a ( k ) converges iff  sum_ n  e. 
NN0 2 ^ n  x.  a ( 2 ^ n ) converges. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  NN )  ->  0  <_  ( F `  k ) )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  n  e.  NN0 )  ->  ( G `  n )  =  ( (
 2 ^ n )  x.  ( F `  ( 2 ^ n ) ) ) )   =>    |-  ( ph  ->  (  seq  1 (  +  ,  F )  e.  dom  ~~>  <->  seq  0 (  +  ,  G )  e.  dom  ~~>  ) )
 
5.8.6  Miscellaneous converging and diverging sequences
 
Theoremdivrcnv 12259* The sequence of reciprocals of real numbers, multiplied by the factor  A, converges to zero. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( A  e.  CC  ->  ( n  e.  RR+  |->  ( A  /  n ) )  ~~> r  0 )
 
Theoremdivcnv 12260* The sequence of reciprocals of natural numbers, multiplied by the factor  A, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 18-Sep-2014.)
 |-  ( A  e.  CC  ->  ( n  e.  NN  |->  ( A  /  n ) )  ~~>  0 )
 
Theoremflo1 12261 The floor function satisfies  |_ ( x )  =  x  +  O
( 1 ). (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( x  e.  RR  |->  ( x  -  ( |_ `  x ) ) )  e.  O ( 1 )
 
Theoremsupcvg 12262* Extract a sequence  f in  X such that the image of the points in the bounded set  A converges to the supremum  S of the set. Similar to Equation 4 of [Kreyszig] p. 144. The proof uses countable choice ax-cc 8015. (Contributed by Mario Carneiro, 15-Feb-2013.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
 |-  X  e.  _V   &    |-  S  =  sup ( A ,  RR ,  <  )   &    |-  R  =  ( n  e.  NN  |->  ( S  -  (
 1  /  n )
 ) )   &    |-  ( ph  ->  X  =/=  (/) )   &    |-  ( ph  ->  F : X -onto-> A )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   =>    |-  ( ph  ->  E. f
 ( f : NN --> X  /\  ( F  o.  f )  ~~>  S ) )
 
Theoreminfcvgaux1i 12263* Auxiliary theorem for applications of supcvg 12262. Hypothesis for several supremum theorems. (Contributed by NM, 8-Feb-2008.)
 |-  R  =  { x  |  E. y  e.  X  x  =  -u A }   &    |-  (
 y  e.  X  ->  A  e.  RR )   &    |-  Z  e.  X   &    |-  E. z  e. 
 RR  A. w  e.  R  w  <_  z   =>    |-  ( R  C_  RR  /\  R  =/=  (/)  /\  E. z  e.  RR  A. w  e.  R  w  <_  z
 )
 
Theoreminfcvgaux2i 12264* Auxiliary theorem for applications of supcvg 12262. (Contributed by NM, 4-Mar-2008.)
 |-  R  =  { x  |  E. y  e.  X  x  =  -u A }   &    |-  (
 y  e.  X  ->  A  e.  RR )   &    |-  Z  e.  X   &    |-  E. z  e. 
 RR  A. w  e.  R  w  <_  z   &    |-  S  =  -u sup ( R ,  RR ,  <  )   &    |-  ( y  =  C  ->  A  =  B )   =>    |-  ( C  e.  X  ->  S  <_  B )
 
Theoremharmonic 12265 The harmonic series  H diverges. This fact follows from the stronger emcl 20244, which establishes that the harmonic series grows as  log n  +  gamma  + o(1), but this uses a more elementary method, attributed to Nicole Oresme (1323-1382). (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( 1 
 /  n ) )   &    |-  H  =  seq  1 (  +  ,  F )   =>    |-  -.  H  e.  dom  ~~>
 
5.8.7  Arithmetic series
 
Theoremarisum 12266* Arithmetic series sum of the first 
N positive integers. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  sum_ k  e.  ( 1
 ... N ) k  =  ( ( ( N ^ 2 )  +  N )  / 
 2 ) )
 
Theoremarisum2 12267* Arithmetic series sum of the first 
N nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0
 ... ( N  -  1 ) ) k  =  ( ( ( N ^ 2 )  -  N )  / 
 2 ) )
 
Theoremtrireciplem 12268 Lemma for trirecip 12269. Show that the sum converges. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  F  =  ( n  e.  NN  |->  ( 1 
 /  ( n  x.  ( n  +  1
 ) ) ) )   =>    |-  seq  1 (  +  ,  F )  ~~>  1
 
Theoremtrirecip 12269 The sum of the reciprocals of the triangle numbers converge to two. (Contributed by Scott Fenton, 23-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |- 
 sum_ k  e.  NN  ( 2  /  (
 k  x.  ( k  +  1 ) ) )  =  2
 
5.8.8  Geometric series
 
Theoremexpcnv 12270* A sequence of powers of a complex number  A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   =>    |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
 
Theoremexplecnv 12271* A sequence of terms converges to zero when it is less than powers of a number  A whose absolute value is smaller than 1. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  <_  ( A ^ k ) )   =>    |-  ( ph  ->  F  ~~>  0 )
 
Theoremgeoserg 12272* The value of the finite geometric series  A ^ M  +  A ^ ( M  + 
1 )  +...  +  A ^
( N  -  1 ). (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  1 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   =>    |-  ( ph  ->  sum_ k  e.  ( M..^ N ) ( A ^ k
 )  =  ( ( ( A ^ M )  -  ( A ^ N ) )  /  ( 1  -  A ) ) )
 
Theoremgeoser 12273* The value of the finite geometric series  1  +  A ^
1  +  A ^
2  +...  +  A ^
( N  -  1 ). (Contributed by NM, 12-May-2006.) (Proof shortened by Mario Carneiro, 15-Jun-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  1 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
 )  =  ( ( 1  -  ( A ^ N ) ) 
 /  ( 1  -  A ) ) )
 
Theoremgeolim 12274* The partial sums in the infinite series  1  +  A ^
1  +  A ^
2... converge to  ( 1  /  (
1  -  A ) ). (Contributed by NM, 15-May-2006.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( F `  k
 )  =  ( A ^ k ) )   =>    |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  (
 1  -  A ) ) )
 
Theoremgeolim2 12275* The partial sums in the geometric series  A ^ M  +  A ^ ( M  + 
1 )... converge to  ( ( A ^ M )  / 
( 1  -  A
) ). (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <  1 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  ->  ( F `  k )  =  ( A ^
 k ) )   =>    |-  ( ph  ->  seq 
 M (  +  ,  F )  ~~>  ( ( A ^ M )  /  ( 1  -  A ) ) )
 
Theoremgeoreclim 12276* The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  1  <  ( abs `  A ) )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( F `  k
 )  =  ( ( 1  /  A ) ^ k ) )   =>    |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( A  /  ( A  -  1 ) ) )
 
Theoremgeo2sum 12277* The value of the finite geometric series  2 ^ -u 1  +  2 ^ -u 2  +...  +  2 ^
-u N, multiplied by a constant. (Contributed by Mario Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  CC )  ->  sum_ k  e.  (
 1 ... N ) ( A  /  ( 2 ^ k ) )  =  ( A  -  ( A  /  (
 2 ^ N ) ) ) )
 
Theoremgeo2sum2 12278* The value of the finite geometric series  1  +  2  + 
4  +  8  +...  +  2 ^ ( N  -  1 ). (Contributed by Mario Carneiro, 7-Sep-2016.)
 |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0..^ N ) ( 2 ^ k )  =  ( ( 2 ^ N )  -  1
 ) )
 
Theoremgeo2lim 12279* The value of the infinite geometric series  2 ^ -u 1  +  2 ^ -u 2  +... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  F  =  ( k  e.  NN  |->  ( A 
 /  ( 2 ^
 k ) ) )   =>    |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  ~~>  A )
 
Theoremgeomulcvg 12280* The geometric series converges even if it is multiplied by  k to result in the larger series  k  x.  A ^
k. (Contributed by Mario Carneiro, 27-Mar-2015.)
 |-  F  =  ( k  e.  NN0  |->  ( k  x.  ( A ^
 k ) ) )   =>    |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  ,  F )  e.  dom  ~~>  )
 
Theoremgeoisum 12281* The infinite sum of  1  +  A ^ 1  +  A ^ 2... is  ( 1  /  ( 1  -  A ) ). (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  sum_ k  e.  NN0  ( A ^ k )  =  ( 1  /  (
 1  -  A ) ) )
 
Theoremgeoisumr 12282* The infinite sum of reciprocals  1  +  ( 1  /  A ) ^ 1  +  ( 1  /  A ) ^ 2... is  A  / 
( A  -  1 ). (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  1  <  ( abs `  A ) ) 
 ->  sum_ k  e.  NN0  ( ( 1  /  A ) ^ k
 )  =  ( A 
 /  ( A  -  1 ) ) )
 
Theoremgeoisum1 12283* The infinite sum of  A ^ 1  +  A ^ 2... is  ( A  /  ( 1  -  A ) ). (Contributed by NM, 1-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  sum_ k  e.  NN  ( A ^ k )  =  ( A  /  (
 1  -  A ) ) )
 
Theoremgeoisum1c 12284* The infinite sum of  A  x.  ( R ^ 1 )  +  A  x.  ( R ^ 2 )... is  ( A  x.  R )  /  (
1  -  R ). (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( A  x.  ( R ^
 k ) )  =  ( ( A  x.  R )  /  (
 1  -  R ) ) )
 
Theorem0.999... 12285 The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e.  9  /  10 ^
1  +  9  /  10 ^ 2  +  9  /  10 ^ 3  +  ..., is exactly equal to 1, according to ZF set theory. Interestingly, about 40% of the people responding to a poll at http://forum.physorg.com/index.php?showtopic=13177 disagree. (Contributed by NM, 2-Nov-2007.)
 |- 
 sum_ k  e.  NN  ( 9  /  ( 10 ^ k ) )  =  1
 
Theoremgeoihalfsum 12286 Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 12283. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 12285 proves a similar claim for .999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |- 
 sum_ k  e.  NN  ( 1  /  (
 2 ^ k ) )  =  1
 
5.8.9  Ratio test for infinite series convergence
 
Theoremcvgrat 12287* Ratio test for convergence of a complex infinite series. If the ratio  A of the absolute values of of successive terms in an infinite sequence  F is less than 1 for all terms beyond some index  B, then the infinite sum of the terms of  F converges to a complex number. Equivalent to first part of Exercise 4 of [Gleason] p. 182. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 27-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  1 )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  W ) 
 ->  ( abs `  ( F `  ( k  +  1 ) ) ) 
 <_  ( A  x.  ( abs `  ( F `  k ) ) ) )   =>    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
 
5.8.10  Mertens' theorem
 
Theoremmertenslem1 12288* Lemma for mertens 12290. (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  ( ( ph  /\  j  e.  NN0 )  ->  ( F `  j )  =  A )   &    |-  ( ( ph  /\  j  e.  NN0 )  ->  ( K `  j
 )  =  ( abs `  A ) )   &    |-  (
 ( ph  /\  j  e. 
 NN0 )  ->  A  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( G `  k
 )  =  B )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( H `  k
 )  =  sum_ j  e.  ( 0 ... k
 ) ( A  x.  ( G `  ( k  -  j ) ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  K )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq  0 (  +  ,  G )  e.  dom  ~~>  )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  T  =  {
 z  |  E. n  e.  ( 0 ... (
 s  -  1 ) ) z  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
 ) ) }   &    |-  ( ps 
 <->  ( s  e.  NN  /\ 
 A. n  e.  ( ZZ>=
 `  s ) ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
 ) )  <  (
 ( E  /  2
 )  /  ( sum_ j  e.  NN0  ( K `  j )  +  1 ) ) ) )   &    |-  ( ph  ->  ( ps  /\  ( t  e.  NN0  /\ 
 A. m  e.  ( ZZ>=
 `  t ) ( K `  m )  <  ( ( ( E  /  2 ) 
 /  s )  /  ( sup ( T ,  RR ,  <  )  +  1 ) ) ) ) )   &    |-  ( ph  ->  ( 0  <_  sup ( T ,  RR ,  <  ) 
 /\  ( T  C_  RR  /\  T  =/=  (/)  /\  E. z  e.  RR  A. w  e.  T  w  <_  z
 ) ) )   =>    |-  ( ph  ->  E. y  e.  NN0  A. m  e.  ( ZZ>= `  y )
 ( abs `  sum_ j  e.  ( 0 ... m ) ( A  x.  sum_
 k  e.  ( ZZ>= `  ( ( m  -  j )  +  1
 ) ) B ) )  <  E )
 
Theoremmertenslem2 12289* Lemma for mertens 12290. (Contributed by Mario Carneiro, 28-Apr-2014.)
 |-  ( ( ph  /\  j  e.  NN0 )  ->  ( F `  j )  =  A )   &    |-  ( ( ph  /\  j  e.  NN0 )  ->  ( K `  j
 )  =  ( abs `  A ) )   &    |-  (
 ( ph  /\  j  e. 
 NN0 )  ->  A  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( G `  k
 )  =  B )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( H `  k
 )  =  sum_ j  e.  ( 0 ... k
 ) ( A  x.  ( G `  ( k  -  j ) ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  K )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq  0 (  +  ,  G )  e.  dom  ~~>  )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  T  =  {
 z  |  E. n  e.  ( 0 ... (
 s  -  1 ) ) z  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
 ) ) }   &    |-  ( ps 
 <->  ( s  e.  NN  /\ 
 A. n  e.  ( ZZ>=
 `  s ) ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
 ) )  <  (
 ( E  /  2
 )  /  ( sum_ j  e.  NN0  ( K `  j )  +  1 ) ) ) )   =>    |-  ( ph  ->  E. y  e.  NN0  A. m  e.  ( ZZ>=
 `  y ) ( abs `  sum_ j  e.  ( 0 ... m ) ( A  x.  sum_
 k  e.  ( ZZ>= `  ( ( m  -  j )  +  1
 ) ) B ) )  <  E )
 
Theoremmertens 12290* Mertens' thoerem. If  A ( j ) is an absolutely convergent series and  B ( k ) is convergent, then  ( sum_ j  e.  NN0 A ( j )  x.  sum_ k  e.  NN0 B ( k ) )  =  sum_ k  e. 
NN0 sum_ j  e.  ( 0 ... k ) ( A ( j )  x.  B ( k  -  j ) ) (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  ( ( ph  /\  j  e.  NN0 )  ->  ( F `  j )  =  A )   &    |-  ( ( ph  /\  j  e.  NN0 )  ->  ( K `  j
 )  =  ( abs `  A ) )   &    |-  (
 ( ph  /\  j  e. 
 NN0 )  ->  A  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( G `  k
 )  =  B )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  NN0 )  ->  ( H `  k
 )  =  sum_ j  e.  ( 0 ... k
 ) ( A  x.  ( G `  ( k  -  j ) ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  K )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq  0 (  +  ,  G )  e.  dom  ~~>  )   =>    |-  ( ph  ->  seq  0
 (  +  ,  H ) 
 ~~>  ( sum_ j  e.  NN0  A  x.  sum_ k  e.  NN0  B ) )
 
5.9  Elementary trigonometry
 
5.9.1  The exponential, sine, and cosine functions
 
Syntaxce 12291 Extend class notation to include the exponential function.
 class  exp
 
Syntaxceu 12292 Extend class notation to include Euler's constant = 2.7182818....
 class  _e
 
Syntaxcsin 12293 Extend class notation to include the sine function.
 class  sin
 
Syntaxccos 12294 Extend class notation to include the cosine function.
 class  cos
 
Syntaxctan 12295 Extend class notation to include the tangent function.
 class  tan
 
Syntaxcpi 12296 Extend class notation to include pi = 3.14159....
 class  pi
 
Definitiondf-ef 12297* Define the exponential function. (Contributed by NM, 14-Mar-2005.)
 |- 
 exp  =  ( x  e.  CC  |->  sum_ k  e.  NN0  ( ( x ^
 k )  /  ( ! `  k ) ) )
 
Definitiondf-e 12298 Define Euler's constant 2.71828.... (Contributed by NM, 14-Mar-2005.)
 |-  _e  =  ( exp `  1 )
 
Definitiondf-sin 12299 Define the sine function. (Contributed by NM, 14-Mar-2005.)
 |- 
 sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) ) 
 /  ( 2  x.  _i ) ) )
 
Definitiondf-cos 12300 Define the cosine function. (Contributed by NM, 14-Mar-2005.)
 |- 
 cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x ) )  +  ( exp `  ( -u _i  x.  x ) ) ) 
 /  2 ) )
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