I need to evaluate a sum
$\sum\limits_{m=0}^\infty f\left(\frac{m}{L}\right)$
and tried to approximate it by an integral for $L\rightarrow\infty$
$\sum\limits_{m=0}^\infty f\left(\frac{m}{L}\right)\approx L\int\limits_0^\infty f(x) dx$
where I will be left with some error $\propto L^{-\alpha}$ for some $\alpha >0$. Everything is finite and converges nicely, however the approximation I made seems too course for my needs, so I was wondering:
Is there a better approximation to improve the convergence, or are there error terms to some order that are worth including?