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