How do I prove that $\sum_ {k=0}^ {\infty} \frac{\sin (kx)^2}{ (1+k^ 2x^2)}$ is not continuous at zero?

Question

How do I prove that $\sum_ {k=0}^ {\infty} \frac{\sin (kx)^2}{ (1+k^ 2x^2)}$ is not continuous at zero?

This is a question on uniform convergence I tried to solve it I just don't have any clue how I can approach this problem. Can you give me any hints?

Answer 1

Let us call $f(x):=\sum_{k=0}^{+\infty}\frac{\sin^2(kx)}{1+k^2x^2}$. Note that $f(0)=0$ and that continuity would imply $f(x_n)\to 0$ as $n\to +\infty$ for every sequence $(x_n)_n$ converging to $0$. However you can see that the choice $x_n=\frac{\pi}{2n}$ gives $f(x_n)\geq\frac{1}{1+\frac{\pi^2}{4}}$ for every $n$, thus $f(x_n)$ cannot converge to $0$.