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.
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.
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.