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