Suppose $\sum_{n=1}^{\infty}f(n)$ converges.
$0 \le f(x)$ on [1,$\infty$] and $\int_{1}^{N}f(x)dx < \sum_{n=1}^{\infty}f(n) < \infty$.
Define a sequence $a_N=\int_{1}^{N}f(x)dx$ then $a_N$ is bounded above and increasing,
therefore converges.
In here, I want to conclude $\int_{1}^{\infty}f(x)dx$ also converges.
It is possible that just applying limit and saying it is true for $\infty$?
Or is there any other theorem that I can use?