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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisumcl 12101* The sum of a converging infinite series is a complex number. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 23-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  e.  CC )
 
Theoremisummulc2 12102* An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-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  ->  B  e.  CC )   =>    |-  ( ph  ->  ( B  x.  sum_ k  e.  Z  A )  = 
 sum_ k  e.  Z  ( B  x.  A ) )
 
Theoremisummulc1 12103* An infinite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 23-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  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 sum_ k  e.  Z  A  x.  B )  = 
 sum_ k  e.  Z  ( A  x.  B ) )
 
Theoremisumdivc 12104* An infinite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-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  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( sum_ k  e.  Z  A  /  B )  =  sum_ k  e.  Z  ( A 
 /  B ) )
 
Theoremisumrecl 12105* The sum of a converging infinite real series is a real number. (Contributed 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  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  A  e.  RR )
 
Theoremisumge0 12106* An infinite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 28-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  RR )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  A )   =>    |-  ( ph  ->  0  <_  sum_
 k  e.  Z  A )
 
Theoremisumadd 12107* Addition of infinite sums. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq 
 M (  +  ,  G )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  ( A  +  B )  =  ( sum_ k  e.  Z  A  +  sum_ k  e.  Z  B ) )
 
Theoremsumsplit 12108* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  ( A  u.  B )  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  C , 
 0 ) )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  if ( k  e.  B ,  C , 
 0 ) )   &    |-  (
 ( ph  /\  k  e.  ( A  u.  B ) )  ->  C  e.  CC )   &    |-  ( ph  ->  seq 
 M (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ph  ->  seq  M (  +  ,  G )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  ( A  u.  B ) C  =  ( sum_ k  e.  A  C  +  sum_ k  e.  B  C ) )
 
Theoremfsump1i 12109* Optimized version of fsump1 12096 for making sums of a concrete number of terms. (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  N  =  ( K  +  1 )   &    |-  ( k  =  N  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  Z )  ->  A  e.  CC )   &    |-  ( ph  ->  ( K  e.  Z  /\  sum_
 k  e.  ( M
 ... K ) A  =  S ) )   &    |-  ( ph  ->  ( S  +  B )  =  T )   =>    |-  ( ph  ->  ( N  e.  Z  /\  sum_
 k  e.  ( M
 ... N ) A  =  T ) )
 
Theoremfsum2dlem 12110* Lemma for fsum2d 12111- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  ( z  =  <. j ,  k >.  ->  D  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  B  e.  Fin )   &    |-  ( ( ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  C  e.  CC )   &    |-  ( ph  ->  -.  y  e.  x )   &    |-  ( ph  ->  ( x  u.  { y } )  C_  A )   &    |-  ( ps  <->  sum_ j  e.  x  sum_
 k  e.  B  C  =  sum_ z  e.  U_  j  e.  x  ( { j }  X.  B ) D )   =>    |-  ( ( ph  /\  ps )  ->  sum_ j  e.  ( x  u.  { y }
 ) sum_ k  e.  B  C  =  sum_ z  e.  U_  j  e.  ( x  u.  { y }
 ) ( { j }  X.  B ) D )
 
Theoremfsum2d 12111* Write a double sum as a sum over a two-dimensional region. Note that  B ( j ) is a function of  j. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( z  =  <. j ,  k >.  ->  D  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  B  e.  Fin )   &    |-  ( ( ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_ k  e.  B  C  =  sum_ z  e.  U_  j  e.  A  ( { j }  X.  B ) D )
 
Theoremfsumxp 12112* Combine two sums into a single sum over the cartesian product. (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  ( z  =  <. j ,  k >.  ->  D  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ( ph  /\  (
 j  e.  A  /\  k  e.  B )
 )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_ k  e.  B  C  =  sum_ z  e.  ( A  X.  B ) D )
 
Theoremfsumcnv 12113* Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  ( x  =  <. j ,  k >.  ->  B  =  D )   &    |-  ( y  = 
 <. k ,  j >.  ->  C  =  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  Rel  A )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ x  e.  A  B  =  sum_ y  e.  `'  A C )
 
Theoremfsumcom2 12114* Interchange order of summation. Note that  B ( j ) and  D
( k ) are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  B  e.  Fin )   &    |-  ( ph  ->  ( ( j  e.  A  /\  k  e.  B ) 
 <->  ( k  e.  C  /\  j  e.  D ) ) )   &    |-  (
 ( ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  E  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_
 k  e.  B  E  =  sum_ k  e.  C  sum_
 j  e.  D  E )
 
Theoremfsumcom 12115* Interchange order of summation. (Contributed by NM, 15-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  (
 ( ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_
 k  e.  B  C  =  sum_ k  e.  B  sum_
 j  e.  A  C )
 
Theoremfsum0diaglem 12116* Lemma for fsum0diag 12117. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
 |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  (
 0 ... ( N  -  j ) ) ) 
 ->  ( k  e.  (
 0 ... N )  /\  j  e.  ( 0 ... ( N  -  k
 ) ) ) )
 
Theoremfsum0diag 12117* Two ways to express "the sum of  A ( j ,  k ) over the triangular region  M  <_  j,  M  <_  k,  j  +  k  <_  N." (Contributed by NM, 31-Dec-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
 |-  ( ( ph  /\  (
 j  e.  ( 0
 ... N )  /\  k  e.  ( 0 ... ( N  -  j
 ) ) ) ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  (
 0 ... N ) sum_ k  e.  ( 0 ... ( N  -  j
 ) ) A  =  sum_
 k  e.  ( 0
 ... N ) sum_ j  e.  ( 0 ... ( N  -  k
 ) ) A )
 
Theoremfsumrev 12118* Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( j  =  ( K  -  k
 )  ->  A  =  B )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( K  -  N ) ... ( K  -  M ) ) B )
 
Theoremfsumshft 12119* Index shift of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( j  =  ( k  -  K )  ->  A  =  B )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  +  K ) ... ( N  +  K ) ) B )
 
Theoremfsumshftm 12120* Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( j  =  ( k  +  K )  ->  A  =  B )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  -  K ) ... ( N  -  K ) ) B )
 
Theoremfsumrev2 12121* Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 13-Apr-2016.)
 |-  ( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( j  =  ( ( M  +  N )  -  k
 )  ->  A  =  B )   =>    |-  ( ph  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( M ... N ) B )
 
Theoremfsum0diag2 12122* Two ways to express "the sum of  A ( j ,  k ) over the triangular region  0  <_  j, 
0  <_  k,  j  +  k  <_  N." (Contributed by Mario Carneiro, 21-Jul-2014.)
 |-  ( x  =  k 
 ->  B  =  A )   &    |-  ( x  =  (
 k  -  j ) 
 ->  B  =  C )   &    |-  ( ( ph  /\  (
 j  e.  ( 0
 ... N )  /\  k  e.  ( 0 ... ( N  -  j
 ) ) ) ) 
 ->  A  e.  CC )   =>    |-  ( ph  ->  sum_ j  e.  (
 0 ... N ) sum_ k  e.  ( 0 ... ( N  -  j
 ) ) A  =  sum_
 k  e.  ( 0
 ... N ) sum_ j  e.  ( 0 ... k ) C )
 
Theoremfsummulc2 12123* A finite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  ( C  x.  sum_ k  e.  A  B )  =  sum_ k  e.  A  ( C  x.  B ) )
 
Theoremfsummulc1 12124* A finite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  ( sum_ k  e.  A  B  x.  C )  =  sum_ k  e.  A  ( B  x.  C ) )
 
Theoremfsumdivc 12125* A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 sum_ k  e.  A  B  /  C )  = 
 sum_ k  e.  A  ( B  /  C ) )
 
Theoremfsumneg 12126* Negation of a finite sum. (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  ->  sum_ k  e.  A  -u B  =  -u sum_ k  e.  A  B )
 
Theoremfsumsub 12127* 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 12128* 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 12129* 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 12130* 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 12131* 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 12132* 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 12133* 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 12134* 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 12135* 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 12136* 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 12137* 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 12138* 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 12139* 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 12140* 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 12141* 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 12142* Lemma for fsumre 12143, fsumim 12144, and fsumcj 12145. (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 12143* 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 12144* 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 12145* 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 12146* 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 12147* 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 12148* 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 12149* 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 12150* 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 12151* 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 12152* 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 12153* 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 12154* 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 12155* 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 12156* 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 12157* 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 12158* 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 12159* 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 12160* 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 12161* 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 12162* 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 12163* Translate the Ackermann bijection ackbij1 7748 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 12164* Lemma for binom 12165 (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 12165* 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 12164. (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 12166* 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 12167* 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 12168* 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 12169 Lemma for bcxmas 12171. (Contributed by Paul Chapman, 18-May-2007.)
 |-  ( A  =  B  ->  ( ( N  +  A )  _C  A )  =  ( ( N  +  B )  _C  B ) )
 
Theorembcxmaslem2 12170 Lemma for bcxmas 12171. (Contributed by Paul Chapman, 18-May-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C )
 )  =  ( ( A  +  C )  +  B ) )
 
Theorembcxmas 12171* 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 12172* 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 12173* 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 12174* 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 12175* 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 12176* 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 12177* 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 12178* 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 12179* 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 12180* 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 12181* 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 12182* Lemma for climcnds 12184: 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 12183* Lemma for climcnds 12184: 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 12184* 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 12185* 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 12186* 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 12187 The floor function satisfies  |_ ( x )  =  x  +  O
( 1 ). (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( x  e.  RR  |->  ( x  -  ( |_ `  x ) ) )  e.  O ( 1 )
 
Theoremsupcvg 12188* 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 7945. (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 12189* Auxiliary theorem for applications of supcvg 12188. 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 12190* Auxiliary theorem for applications of supcvg 12188. (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 12191 The harmonic series  H diverges. This fact follows from the stronger emcl 20128, 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 12192* 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 12193* 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 12194 Lemma for trirecip 12195. 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 12195 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 12196* 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 12197* 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 12198* 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 12199* 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 12200* 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 ) ) )
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