Calculation of $\sum xf(x)$ given $\sum f(x)$?

Question

Given the sum $\sum f(x)$, is it possible to calculate $\sum xf(x)$? What if it is constrained that these are infinite sums? I assume that this doesn't exist because then computing variance would be very easy, and I've heard that it isn't.

Answer 1

Given $f(1)+f(2)$ is it possible to compute $f(1)+2f(2)$? Even then, the answer is "no".

Another example. A discrete probability distribution satisfies $\sum f(x) = 1$. The expectation of that distribution is $\sum x f(x)$. But of course not all discrete probability distributions have the same expectation.