Let $f:[1, \infty)\rightarrow \mathbb R$. If $\lim_{x\to \infty}f(x)=0$ and $\lim_{n\to \infty} \int_{1}^{n}f$, $n\in \mathbb N$ exists then $\int_{1}^{\infty} f$ exists
I´m having a really hard time with this problem. I haven´t been able to prove it or give a counterexample.
I would really appreciate any ideas or suggestions.