Moving an integral to the inside of a sum

Question

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) $$ \frac{1}{2\pi} \int_{-\pi}^{\pi}\sum_{-\infty}^{\infty}a_n e^{i(n-m)x} \; dx = \frac{1}{2\pi} \sum_{-\infty}^{\infty} \int_{-\pi}^{\pi}a_n e^{i(n-m)x} \; dx$$

Answer 1

If the series $\sum_{-\infty}^{\infty}a_n e^{i(n-m)x}$ is uniformly convergent on $[- \pi , \pi]$, then you can move the integral from the outside of a sum to inside the sum.