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## Homework Statement

Hi all.

There is an example in my book, where we have the following Fourier series, and the author writes it as:

[tex]

f(x) = \sum\limits_{n = 1}^\infty {\left( {\frac{{( - 1)^n }}{{n^2 }}\cos \left( {\frac{{n\pi x}}{p}} \right) + \frac{1}{n}\sin \left( {\frac{{n\pi x}}{p}} \right)} \right)} = \sum\limits_{n = 1}^\infty {\frac{{( - 1)^n }}{{n^2 }}\cos \left( {\frac{{n\pi x}}{p}} \right)} + \sum\limits_{n = 1}^\infty {\frac{1}{n}\sin \left( {\frac{{n\pi x}}{p}} \right)}

[/tex]

First of all, we are only allowed to split the sum up if, and only if each part in the summation converges, but the author does not check if they do or don't. Is he making a mistake or am I missing something?