I am deriving the fourier coefficient formula, and was wondering under what conditions I can move an integral from the outside of a sum to inside the sum? (as I have done below) $$ \frac{1}{2\pi} \int_{-\pi}^{\pi}\sum_{-\infty}^{\infty}a_n e^{i(n-m)x} \; dx = \frac{1}{2\pi} \sum_{-\infty}^{\infty} \int_{-\pi}^{\pi}a_n e^{i(n-m)x} \; dx$$
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Have you searched on this site ? https://math.stackexchange.com/questions/83721/when-can-a-sum-and-integral-be-interchanged – InfiniteLooper Nov 16 '20 at 11:46
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$\sum |a_n|^{2} <\infty$ is sufficient. – Kavi Rama Murthy Nov 16 '20 at 11:53