TITLE: A tricky sum to infinity
QUESTION [12 upvotes]: I try to solve the following tricky limit:
$$\lim_{x\rightarrow\infty} \sum_{k=1}^{\infty} \frac{kx}{(k^2+x)^2} $$
For some large values, W|A shows that its limit tends to $\frac{1}{2}$ but not sure how to prove that.

REPLY [16 votes]: ETA: These bounds are wrong, as $\frac{kx}{(k^2+x)^2}$ is not monotone in $k$. For a fixed version of this answer, see robjohn's answer here.

Notice that, for fixed $x$, your sum is less than
$$\int_0^\infty \frac{kx}{(k^2+x)^2} \, dk=\frac{1}{2}\, ,$$
and greater than
$$\int_1^\infty \frac{kx}{(k^2+x)^2} \, dk=\frac{x}{2(1+x)} \, ,$$
and then apply the squeeze theorem.