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I need to prove that $$f(x)= \sum_{n=1}^{\infty} \frac{\sin nx + \cos nx }{n^3}$$ is well defined on $\mathbb{R}$, is a differentiable function, and it's derivative is differentiable continuous.

I used Weierstrass's M-Test and proved that the series uniformly converges (since it is smaller than $\sum_{0}^{\infty}\frac{2}{n^3}$- Is it enough?) so it's differentiable, How do I check wheter $f$ continuous or not?

$x$ is all the real line.

Thanks a lot!

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up vote 3 down vote accepted

Your application of the M-test is correct and proves that the sum converges uniformly on the real line. Since the limit of a uniformly convergent sequence of continuous functions is also continuous, this proves $f$ is continuous immediately.

To show $f$ is differentiable use the following theorem: If $f_n : \mathbb{R} \to \mathbb{R} $ is a sequence of continuously differentiable functions, uniformly convergent to $f$ and the sequence $f'_n$ converges uniformly to $g$, then $f$ is differentiable and $f'=g.$ With this theorem and a similar application of the M-test again, we can establish that our sum $f$ is continuously differentiable and $$ f'(x) = \sum_{n=0}^{\infty} \frac{ \cos nx - \sin nx }{n^2} .$$

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