Assume f(x) positive and continuous,for improper integrals of the first kind, is there an analog of the n-th term divergence test for series ? It would say:

$\displaystyle \int_{a}^{\infty} f(x)dx\ converges \Rightarrow \lim_{x \to +\infty} f(x) = 0. $

Give a proof or counterexample.

I think $\displaystyle \lim_{x \to +\infty} f(x) \neq 0$,but I can not give a counterexample.