Does the series $ \ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^2+n^2}{n^3} $ converges uniformly on $ \ \ \ [-1,1] \ $?

Question

Does the series $ \ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^2+n^2}{n^3} $ converges uniformly on $ \ \ \ [-1,1] \ $?

Answer:

$ \ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^2+n^2}{n^3} \\ =\sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^2}{n^3}+ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n} \\ \leq \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^3}+ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n} , \ (\because |x| \leq 1 ) \\ = A+B $

Since both the series $ \ A,B \ $ are alternating series , both converges uniformly on $ \ [-1,1] \ $.

So the given series converges on $ \ [-1,1] \ $ uniformly.

But I am not sure of my work.

Please someone verify my work .

Answer 1

This is not correct. Why would those inequalities be true?

You can prove that the series converges uniformly using Dirichlet's test:

1. $\left(\sum_{n=1}^N(-1)^n\right)_{N\in\mathbb N}$ is bounded;
2. $\left(\frac{x^2+n^2}{n^3}\right)_{n\in\mathbb N}$ is monotonic for each $x\in[-1,1]$;
3. $\left(\frac{x^2+n^2}{n^3}\right)_{n\in\mathbb N}$ converges uniformly to the null function.