If the sequence is indexed starting at , then we may formally define . The previous formula becomes
A common way to apply Abel's summation formula is to take the limit of one of these formulas as . The resulting formulas are
These equations hold whenever both limits on the right-hand side exist and are finite.
A particularly useful case is the sequence for all . In this case, . For this sequence, Abel's summation formula simplifies to
Similarly, for the sequence and for all , the formula becomes
Upon taking the limit as , we find
assuming that both terms on the right-hand side exist and are finite.
Abel's summation formula can be generalized to the case where is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral:
By taking to be the partial sum function associated to some sequence, this leads to the summation by parts formula.
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Other concepts sometimes known by this name are summation by parts and Abel Plana formula In mathematics Abel s summation formula introduced by Niels Henrik Abel is intensively used in analytic number theory and the study of special functions to compute series Contents 1 Formula 1 1 Variations 2 Examples 2 1 Harmonic numbers 2 2 Representation of Riemann s zeta function 2 3 Reciprocal of Riemann zeta function 3 See also 4 ReferencesFormula Edit Wikibooks has a book on the topic of Analytic Number Theory Useful summation formulas Let a n n 0 displaystyle a n n 0 infty be a sequence of real or complex numbers Define the partial sum function A displaystyle A by A t 0 n t a n displaystyle A t sum 0 leq n leq t a n for any real number t displaystyle t Fix real numbers x lt y displaystyle x lt y and let ϕ displaystyle phi be a continuously differentiable function on x y displaystyle x y Then x lt n y a n ϕ n A y ϕ y A x ϕ x x y A u ϕ u d u displaystyle sum x lt n leq y a n phi n A y phi y A x phi x int x y A u phi u du The formula is derived by applying integration by parts for a Riemann Stieltjes integral to the functions A displaystyle A and ϕ displaystyle phi Variations Edit Taking the left endpoint to be 1 displaystyle 1 gives the formula 0 n x a n ϕ n A x ϕ x 0 x A u ϕ u d u displaystyle sum 0 leq n leq x a n phi n A x phi x int 0 x A u phi u du If the sequence a n displaystyle a n is indexed starting at n 1 displaystyle n 1 then we may formally define a 0 0 displaystyle a 0 0 The previous formula becomes 1 n x a n ϕ n A x ϕ x 1 x A u ϕ u d u displaystyle sum 1 leq n leq x a n phi n A x phi x int 1 x A u phi u du A common way to apply Abel s summation formula is to take the limit of one of these formulas as x displaystyle x to infty The resulting formulas are n 0 a n ϕ n lim x A x ϕ x 0 A u ϕ u d u n 1 a n ϕ n lim x A x ϕ x 1 A u ϕ u d u displaystyle begin aligned sum n 0 infty a n phi n amp lim x to infty bigl A x phi x bigr int 0 infty A u phi u du sum n 1 infty a n phi n amp lim x to infty bigl A x phi x bigr int 1 infty A u phi u du end aligned These equations hold whenever both limits on the right hand side exist and are finite A particularly useful case is the sequence a n 1 displaystyle a n 1 for all n 0 displaystyle n geq 0 In this case A x x 1 displaystyle A x lfloor x 1 rfloor For this sequence Abel s summation formula simplifies to 0 n x ϕ n x 1 ϕ x 0 x u 1 ϕ u d u displaystyle sum 0 leq n leq x phi n lfloor x 1 rfloor phi x int 0 x lfloor u 1 rfloor phi u du Similarly for the sequence a 0 0 displaystyle a 0 0 and a n 1 displaystyle a n 1 for all n 1 displaystyle n geq 1 the formula becomes 1 n x ϕ n x ϕ x 1 x u ϕ u d u displaystyle sum 1 leq n leq x phi n lfloor x rfloor phi x int 1 x lfloor u rfloor phi u du Upon taking the limit as x displaystyle x to infty we find n 0 ϕ n lim x x 1 ϕ x 0 u 1 ϕ u d u n 1 ϕ n lim x x ϕ x 1 u ϕ u d u displaystyle begin aligned sum n 0 infty phi n amp lim x to infty bigl lfloor x 1 rfloor phi x bigr int 0 infty lfloor u 1 rfloor phi u du sum n 1 infty phi n amp lim x to infty bigl lfloor x rfloor phi x bigr int 1 infty lfloor u rfloor phi u du end aligned assuming that both terms on the right hand side exist and are finite Abel s summation formula can be generalized to the case where ϕ displaystyle phi is only assumed to be continuous if the integral is interpreted as a Riemann Stieltjes integral x lt n y a n ϕ n A y ϕ y A x ϕ x x y A u d ϕ u displaystyle sum x lt n leq y a n phi n A y phi y A x phi x int x y A u d phi u By taking ϕ displaystyle phi to be the partial sum function associated to some sequence this leads to the summation by parts formula Examples EditHarmonic numbers Edit If a n 1 displaystyle a n 1 for n 1 displaystyle n geq 1 and ϕ x 1 x displaystyle phi x 1 x then A x x displaystyle A x lfloor x rfloor and the formula yields n 1 x 1 n x x 1 x u u 2 d u displaystyle sum n 1 lfloor x rfloor frac 1 n frac lfloor x rfloor x int 1 x frac lfloor u rfloor u 2 du The left hand side is the harmonic number H x displaystyle H lfloor x rfloor Representation of Riemann s zeta function Edit Fix a complex number s displaystyle s If a n 1 displaystyle a n 1 for n 1 displaystyle n geq 1 and ϕ x x s displaystyle phi x x s then A x x displaystyle A x lfloor x rfloor and the formula becomes n 1 x 1 n s x x s s 1 x u u 1 s d u displaystyle sum n 1 lfloor x rfloor frac 1 n s frac lfloor x rfloor x s s int 1 x frac lfloor u rfloor u 1 s du If ℜ s gt 1 displaystyle Re s gt 1 then the limit as x displaystyle x to infty exists and yields the formula z s s 1 u u 1 s d u displaystyle zeta s s int 1 infty frac lfloor u rfloor u 1 s du This may be used to derive Dirichlet s theorem that z s displaystyle zeta s has a simple pole with residue 1 at s 1 Reciprocal of Riemann zeta function Edit The technique of the previous example may also be applied to other Dirichlet series If a n m n displaystyle a n mu n is the Mobius function and ϕ x x s displaystyle phi x x s then A x M x n x m n displaystyle A x M x sum n leq x mu n is Mertens function and 1 z s n 1 m n n s s 1 M u u 1 s d u displaystyle frac 1 zeta s sum n 1 infty frac mu n n s s int 1 infty frac M u u 1 s du This formula holds for ℜ s gt 1 displaystyle Re s gt 1 See also EditSummation by parts Integration by partsReferences EditApostol Tom 1976 Introduction to Analytic Number Theory Undergraduate Texts in Mathematics Springer Verlag Retrieved from https en wikipedia org w index php title Abel 27s summation formula amp oldid 1083389574, wikipedia, wiki, book, books, library,