Is a sequence of functions differentiable

Question

I have the following series of functions: $$f(x) = \sum_{n=0}^\infty \frac{1}{n^x}$$ I'm trying to show that its differentiable on $(1, \infty)$.

If I differentiate each constituent of the sigma, and the sum of the series (of the derivatives) uniformly converges, can I say the above $f(x)$ is differentiable on $(1, \infty)$?
If this proposition is wrong,can someone give me a hint :(?

Thanks in advance!

Answer 1

Yes. That's part of a standard Real Analysis theorem: if the sequence of functions $(f_n)_{n\in\mathbb N}$ converges pointwise to a function $f$ and if the sequence of functions $(f_n')_{n\in\mathbb N}$ converges uniformly to a function $g$, then $f$ is differentiable and $f'=g$.