If the improper integral $\int_{0}^{\infty}f(x)dx$ is covergent.

Question

Let $f:[0,\infty)\rightarrow [0,\infty)$ be continuous function such that $\int_{0}^{\infty}f(x)<\infty.$ Then which of the following is true?

$a.$ The sequence $\{f(n)\}$ is bounded.

$b. f(n)\rightarrow 0$ as $n\rightarrow\infty.$

$c.$ The series $\sum f(n)$ is convergent.

I can discard second and third and third options by taking particular function $f.$ But i don,t know the general concept. Help me to solve the problem. Thanks in advance.

Answer 1

Hint: Consider tents of height $n$ and base $1/n^3$ (starting from $n=2$).