If a sum is finite, can you conclude the same for the following sum?

Question

If we know that $\sum_{-\infty}^{\infty} f(x) < \infty$,

Then can we conclude that $\sum_{-\infty}^{\infty} x\times f(x) < \infty$?

The answer is not immediately obvious to me. I think there may be a simple counter example, but I've not been able to think of one.

Answer 1

Hint

What do you think if $f(k)=\frac{1}{k^2}$ ?