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
Ifhas a continuous derivativeon the intervalwherethen
(1)
Proof: LetFor integersandinwe have
Summing fromtowe find the first sum telescopes hence
Rearranging this gives
(2)
Integration by parts givesand we can rearrange this to get
(3)
Adding (2) and (3) gives (1).
Examples
Taketo obtain
(4)
Nowso (4) becomeswith
Example
if
Take f(t)=1 over t^s , s>0,{}s neq 1. Euler's summation formula gives
Thereforewhere
IfasandsoifIfand sois also equal toif
Example