Problem:
Let $f\in C^{1}([0,\infty ))$ such that: \int_{1}^{\infty }\left | f^{'}(x) \right |dx converges. The question is to prove the following:
$\left ( \sum_{n=1}^{\infty }f(n) \right )$ converges $\Leftrightarrow \left ( \int_{1}^{\infty }f(x)dx \right )$ converges
I don't know how to prove it. For the direction: $\Leftarrow $ I was trying to use the definition of Rieamann integrals as an infinite sum where the mesh goes to zero, and somehow try to prove that $\left ( \sum_{n=1}^{\infty }f(n) \right )$ converges.
Any solution or ideas for this problem?