Euler's summation formula gives a summation of terms in terms of integral plus some residual terms. It is especially useful in describing the asymptotic behaviour of summations.
Theorem
If
has a continuous derivative
on the interval
where
then
(1)
Proof: Let
For integers
and
in
we have
Summing from
to
we find the first sum telescopes hence
Rearranging this gives
(2)
Integration by parts gives
and we can rearrange this to get
(3)
Adding (2) and (3) gives (1).
Examples
Take
to obtain
(4)
Now
so (4) becomes
with
Example
if
Take f(t)=1 over t^s , s>0,{}s neq 1. Euler's summation formula gives
Therefore
where
If
as
and
so
if
If
and 
so
is also equal to
if
Example
