Consider the function $f(x) = \frac{x}{2}$, defined over the interval $[0, 2\pi]$. Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$.
2026-03-30 07:11:40.1774854700
Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$ using Fourier series
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You just have to expand $y=x/2$ as a Fourier series: $$ {x\over2}=\sin (x)-\frac{1}{2} \sin (2 x)+\frac{1}{3} \sin (3 x)-\frac{1}{4} \sin (4x)+\frac{1}{5} \sin (5 x)+\ldots $$ and put here $x=\pi/2$.