Is this uniformly convergent: $\sum_{k=1}^{\infty} \frac{1}{1+k^2x^2}$ where $x \in (0,1)$?

Question

I am pretty sure that the following is not uniformly convergent: $\sum_{k=1}^{\infty} \frac{1}{1+k^2x^2}$ where $x \in (0,1)$. I am just having trouble coming up with a counterexample.

Any suggestions? Or am I off-base and this is uniformly convergent? If so, can you explain to me why?

Answer 1

Let $f_n(x)$ be the sequence given by

$$f_n(x)=\sum_{k=1}^n\frac{1}{1+k^2x^2}$$

Let $\epsilon=\frac12$. Then, for $x=1/n$ we have

$$|f_n(1/n)|=\sum_{k=1}^n\frac{1}{1+(k/n)^2}\ge \frac{n}{2}>\epsilon$$

Therefore, the convergence is not uniform.