Help with series: a little extra difficulty I refer to this question Help with series: where to start?
but with a little increase of difficulty. I would study the convergence of the series
$$\sum_{n=1}^{+\infty}(-1)^n\frac{nx^3}{n^2+x^2}.$$
Same reasons as in Help with series: where to start?, allow us to conclude that there is no uniform convergence. In particular, it makes no sense to study, for example, the total convergence since total convergence implies the uniforme convergence. The only thing that it makes sense (correct me if I am wrong) is to study the pointwise convergence.
In order to study the pointwise convergence, I use the Leibnitz criterion. I observe that, having
$$f_n(x)=\frac{nx^3}{n^2+x^2},$$
it is:
1)$\;\lim_{n\to +\infty} f_n(x)=0$;
2)$\;f_n(x)\ge 0\iff x\ge 0$;


*$f_n(x)$ is always increasing (when $x>0$),

thus, if $x\ge 0$ the series thus not converge pointwise. My questions are: does my reasoning hold so far?
Moreover, what to say when $x<0$?
I hope someone could help. Thank you in advance!
 A: Your 3 first statements are correct. Your conclusion thus, if $x\ge 0$ the series thus not converge pointwise is not correct.
To study pointwise convergence, you need to fix $n$ and look at the convergence of $\sum_{n=1}^{+\infty}(-1)^n f_n(x)$. This is indeed convergent as we have here an alternating series as the sequence $\{f_n(x)\}$ is eventually decreasing (and positive) and converging to zero. And therefore you can apply (Leibnitz) Alternating series test.
Regarding $x \le 0$, you just have to notice that all the $f_n$ are odd maps.
A: There is a flaw in your argument : for $x\geq 0$, the (non-zero) sequence $(f_n(x))_{n \in \mathbb N}$ cannot be non-negative and go to zero.
For any $x \in \mathbb R$, we have :

*

*$\lim_{n\to \infty} f_n(x) = 0$

*since :
$$f_n(x) = \frac{x^3}{n + \frac{x^2}{n}}$$
the sequence $(f_n(x))_{n\in \mathbb N}$ is either non-negative and (eventually) decreasing (when $x\geq 0$) or non-positive and (eventually)  increasing (when $x\leq 0$)

Therefore, by the Leibniz criterion, $\sum_n (-1)^n f_n(x)$ converges (pointwise) for every $x\in\mathbb R$.
