(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

Full Fourier series: [tex]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) [/tex]

Definition: [tex]\sum_{n=1}^\infty f_n (x) [/tex] converges to [tex]f(x)[/tex] pointwise in [tex](a,b)[/tex] if for each [tex]a<x<b[/tex] we have

[tex]\Big| f(x) - \displaystyle{\sum_{n=1}^\infty f_n (x)} \Big| \to 0 [/tex] as [tex]N\to\infty[/tex].

3. 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?

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# Homework Help: Pointwise convergence of integral of Fourier series

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