Given fourier series:
$$\mathrm{S}\left(x\right) =
{3 \over \pi}\sum_{n = 0}^{\infty}
{\sin\left(\left[2n + 1\right]x\right) \over 2n + 1}\,,\qquad
\left\langle -\pi,\pi\right\rangle
$$
Evaluate: $\displaystyle{%
\int_{-\pi}^{\pi}\mathrm{f}\left(x\right)\,\mathrm{d}x}$.
I suppose that Parseval's identity should be used somehow to calculate this.
I only know how to calculate sum using Parseval, don't know how to apply it to find value of an integral.
Any help would be welcomed.
2026-03-26 14:43:06.1774536186
Calculating integral value of Fourier series
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Why Parseval? It is way easier than that: the integral of an odd integrable function over a symmetric interval with respect to the origin is just zero.