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

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?

• Have you tried evaluating this at $0$? Commented Apr 15, 2022 at 12:17
• @AnCar yeah it's zero i just don't know how to take the limit Commented Apr 15, 2022 at 12:21

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 $$0$$.
• The terms of the sum are all non-negative. Think what happens when you look at the term with $k=n$. Commented Apr 15, 2022 at 12:49