# Derivative of a series of functions

I am given a function expressed as

$$h(x) = \sum_{n=1}^{\infty} f_n(x) = \sum_{n=1}^{\infty} xe^{-n^4x^2}$$

I want to express the derivative $$h'(x), x \in (0,\infty)$$ as a series. From what I understand I need to show three things in order to $$(\sum_{n=1}^{\infty} f_n(x))' = \sum_{n=1}^{\infty} f'_n(x)$$ being true. and those are:

• $$f'_n(x) \in \mathbb{R}, \forall n \in \mathbb{N}$$: Which should be as simple as computing the derivative $$f'_n(x) = e^{-n^4x^2}(1-2n^4x^2)$$ and seeing there is no problem there.

• $$\exists x_0 \in (0,\infty)$$ such that $$\sum_{n=1}^{\infty} f_n(x)$$ converges: which is again simple enough, we can pick i.e. $$x_0 = 1$$, then the series $$\sum_{n=1}^{\infty} e^{-n^4}$$ surely converges.

• $$\sum_{n=1}^{\infty} f'_n(x)$$ converges uniformly for $$x\in (0,\infty)$$: and here is my problem. I know that if $$\sum_{n=1}^{\infty} sup_{x\in(0,\infty)}|f'_n(x)| = \sum_{n=1}^{\infty} sup_{x\in(0,\infty)}|e^{-n^4x^2}(1-2n^4x^2)|$$ converges then $$\sum_{n=1}^{\infty} f'_n(x)$$ converges. To obtain the supremum I computed $$f''_n(x)=0$$ and got $$x\in\{{0,\sqrt{\frac{3}{2n^4}}}\}$$ and the $$lim_{x \rightarrow \infty} f_n(x) = 0$$. So the supremum is for $$x=0$$. Fianlly $$\sum_{n=1}^{\infty} sup_{x\in(0,\infty)}|f'_n(x)| = \sum_{n=1}^{\infty} |f'_n(0)| = \sum_{n=1}^{\infty} 1$$ and this series diverges.

So in the end this doesn't give me any information. What should I do? Am I doing this right? Is the locally uniform convergence good enough in this instance? I believe for $$x\in(\epsilon, \infty)$$ for$$\space small \space \epsilon \in \mathbb{R} \space$$ it would converge, right? If it doesn't converge and the derivative doesn't exist how do I show that?

Thank you.

• Did you mean that $\displaystyle h(x) = \sum_{n=1}^{\infty} f_n(x) = \sum_{n=1}^{\infty} xe^{\color{red}-n^4x^2}$? May 16, 2021 at 22:01
• If $f_n(x)=xe^{n^4x^2}$, then you shouldn't have minus in derivative. May 16, 2021 at 22:02
• Yes sorry, there should have been $-n^4x^2$ in the exponent May 17, 2021 at 7:28