how is this theorem called?

Question

let be the sum

$$ f(\epsilon) = \sum_{n=1}^{\infty}e^{-n\epsilon} $$

• $ f(0)= \infty $ diverges

• for any positive epsilon $ \epsilon >0 $ the sum converges

assume we know the value of $ f(\epsilon) $ as $ \epsilon \to 0 $

then the asymptotics of the partial sum of the coefficients has the asymptotics

$$ \sum_{n=1}^{x}a(n) \sim L^{-1}[f(\epsilon)/\epsilon](x) $$

where $ L^{-1} [f(s)](x) $ is the inverse laplace transform