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[SOLVED] understanding a question Fourier series

dwsmith

Well-known member
Feb 1, 2012
1,673
Apply Theorem 1.4 to evaluate various series of constants.

Theorem 1.4: Let $f$ be periodic and piecewise differentiable. Then at each point $\theta$ the symmetric partial sums
$$
S_N(\theta) = \sum_{n=-N}^Na_ne^{in\theta}
$$
converge to $\frac{1}{2}\left[f(\theta)+f(-\theta)\right]$; if $f$ is continuous at $\theta$, they converge to $f(\theta)$.

So taking this series:
$$
2\sum_{n =1}^{\infty}\left[\frac{1}{2n-1}\sin(2n-1)\theta - \frac{1}{2n}\sin 2n\theta\right]
$$
What am I supposed to do?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,713
Apply Theorem 1.4 to evaluate various series of constants.
That is a fairly vaguely worded problem. It looks as though they want you to see what Theorem 1.4 tells you about the Fourier series for the periodic function defined on the interval $(-\pi,\pi)$ by $f(\theta)=\theta$, by seeing what happens when you choose various values for $\theta.$ I would start by taking $\theta=\pi/2$. Next, look at what happens when $\theta=\pi$ (where the function $f(\theta)$ has a jump discontinuity).