Necessary condition for convergence of improper integral.

Question

If $\int_0^{\infty} f(s)ds$ is convergent , then we can't say this $\lim_{s\rightarrow \infty} f(s) = 0$ if $\lim_{s\rightarrow \infty} f(s)$ doesn't exist, But here if f is nonnegative , then is it also true? I can't find here such counter example when f is nonnegative, please help me!

Answer 1

Let $f(x)=n^{3}(\frac 1 {n^{3}}-|x-n|)$ for $n-\frac 1{n^{3}} < x < n+\frac 1{n^{3}}$ for $n=2,3,...$ and $0$ outside these intervals. Then $f(n)=1$ for all $n$ but $f$ is integrable.