Abel's summation 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. Formula

Let  be a sequence of real or complex numbers. Define the partial sum function  by

for any real number . Fix real numbers , and let  be a continuously differentiable function on . Then:

The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions  and . Variations
Taking the left endpoint to be  gives the formula

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. Examples

Harmonic numbers
If  for  and  then  and the formula yields

The left-hand side is the harmonic number . Representation of Riemann's zeta function
Fix a complex number . If  for  and  then  and the formula becomes

If , then the limit as  exists and yields the formula

where  is the Riemann zeta function. This may be used to derive Dirichlet's theorem that  has a simple pole with residue 1 at . Reciprocal of Riemann zeta function
The technique of the previous example may also be applied to other Dirichlet series. If  is the Möbius function and , then  is Mertens function and

This formula holds for . See also
Summation by parts
Integration by parts

References
 . Number theory
Summability methods