Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question

1 Answer 1

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 $$ f'(x) = \sum_{n=0}^{\infty} \frac{ \cos nx - \sin nx }{n^2} .$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.