Uniform convergence of $\sum_{n=1}^{\infty}x^{n}/n^{2}$

Question

One problem saying:

Is $\sum_{n=1}^{\infty}1/(x^{2}+n^{2})$ uniform convergence?

So I solved it using Weierstrass M test since $\mathbb R $ is complete and $g_k\leq 1/k^2$ and the series consisting of $1/k^2$ is convergent by P-series test.

But what I want to know is the following problem:

Is $\sum_{n=1}^{\infty}x^{n}/n^{2}$ uniformly convergent on $x\in [0,1]$?

To solve it, I tried to use Cauchy Criterion.

So, I tried to find, for given $\epsilon$, certain $N$ for that condition.

For $k\geq N$

$\|g_{k}(x)+\cdots+g_{k+p}(x)\|=\|\frac{x^{k}}{k^{2}}+\cdots+\frac{x^{k+p}}{(k+p)^{2}}\|\leq\|\frac{1}{k^{2}}+\cdots+\frac{1}{k^{2}}\|=\frac{p}{k^{2}}.$

How can I find such $N$?

Answer 1

If $n \geq 1$, then $$ x^{n}/n^{2} \leq 1/n^{2} $$ for all $x \in [0,1]$; by comparison test the uniform convergence follows.