This is in an Electrical Engineering context, where I'm finding power over a period as: $P =\frac{1}{T}\int_{0}^{T} \frac{v(t)^2}{R} dt$, for some arbitrary time $T$ and some constant $R$.
I'm having trouble integrating for $P$ given $v(t)$ as the Fourier series of a square wave: $$v(t) = \frac{4V}{π}•(sin(ωt) + \frac{1}{3}sin(3ωt)+\frac{1}{5}sin(5ωt) ...)$$ for some constant $ω$ and $V$.
My biggest issue is how to integrate the series of $sin$ functions that's also squared.
A square wave of unit amplitude has the Fourier series $$v(t)=\frac{4}{\pi}\sum_{n=0}^{\infty}\frac{1}{2n+1}\sin\left(\frac{2\pi(2n+1)t}{T}\right)$$ so your function $v$ is a square wave of amplitude $V$ and period $T$. This is easy to integrate: $$P=\frac{1}{T}\int_{0}^{T}\frac{v(t)^2}{R}dt=\frac{1}{T}\int_{0}^{T}\frac{V^2}{R}dt=\frac{V^2}{RT}\cdot T=\frac{V^2}{R}.$$