Pointwise convergence of integral of Fourier series

[twizzy]

Homework Statement

If $$f(x)$$ is a piecewise-continuous function in $$[-L,L]$$, show that its indefinite integral $$F(x) = \int_{-L}^x f(s) ds$$ has a full Fourier series that converges pointwise.

Homework Equations

Full Fourier series: $$f(x)=\frac{1}{2}A_0 + \sum_{n=1}^\infty A_n \cos (\frac{n \pi }{L}x) + B_n \sin (\frac{n \pi}{L}x)$$

Definition: $$\sum_{n=1}^\infty f_n (x)$$ converges to $$f(x)$$ pointwise in $$(a,b)$$ if for each $$a<x<b$$ we have
$$\Big| f(x) - \displaystyle{\sum_{n=1}^\infty f_n (x)} \Big| \to 0$$ as $$N\to\infty$$.

The Attempt at a Solution

I think I need to somehow justify integrating term-by-term, but am not sure how to proceed. Any ideas?

 

[ninty]

If you want to integrate term by term, you need uniform convergence.
Haven't really looked at this, so not saying that term by term integration is the solution here.