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.
be a sequence of real or complex numbers.
Define the partial sum function
by for any real number
Fix real numbers
be a continuously differentiable function on
Then: The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions
ϕ
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 this sequence, Abel's summation formula simplifies to Similarly, for the sequence
, 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.
and the formula yields The left-hand side is the harmonic number
Fix a complex number
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 s = 1.
The technique of the previous example may also be applied to other Dirichlet series.
is the Möbius function and
is Mertens function and This formula holds for