Uniform Convergence of a series of functions using the Dirichlet's test

Question

I have recently been trying out some questions on series of functions. I got stuck in one of those problems in which I am supposed to show that the below series of functions is uniformly convergent on any bounded interval.

The series is given by: $$\sum_{1}^\infty (-1)^n\frac{x^2+n}{n^2}$$

I tried using the Dirichlet's test over here by letting $a_n(x)=(-1)^n$ and $b_n(x)=\frac{x^2+n}{n^2}$ but what I am unable to prove here is that $b_n(x)$ is monotonic and uniformly converging to $0$ for all $x$ in a bounded interval.

Please help!

Answer 1

Choose $m\in\Bbb Z^+$ large enough so that your bounded interval is contained in $[-m,m]$. Then show that $\langle b_n:n>m\rangle$ is monotonic and converges uniformly to $0$ on $[-m,m]$. Dirichlet’s test then allows you to conclude that $$\sum_\limits{n>m}(-1)^n\frac{x^2+n}{n^2}$$ converges, which is good enough. It may be helpful to rewrite $b_n$ as

$$b_n=\left(\frac{x}n\right)^2+\frac1n\;.$$

Answer 2

Denote $S_n(x) = \sum_{k=1}^n (-1)^n\frac{x^2+n}{n^2}$ and $s_n(x)=(-1)^n\frac{x^2+n}{n^2}$.

You have

$$\left\vert s_{2n}(x) + s_{2n+1}(x) \right\vert = \left\vert \frac{x^2(4n+1) -2n(2n+1)}{(2n)^2(2n+1)^2} \right\vert \le \frac{M^2 + 1}{2n(2n+1)}$$

for $x \in [0,M]$. Therefore, the series converges uniformly on all bounded intervals as $\sum 1/n^2$ is convergent.

Note: this is not using Dirichlet test but normal convergence.