calculus - Calculating value of the fourier series

In the following fourier series, how the red marked numbers are calculated?

Answer

The easiest way to verify this equality is to work from the right hand side. Expand it into a Fourier series, and check that the result is the series on the left. Then appeal to an appropriate theorem on the pointwise convergence of Fourier series; Dirichlet's theorem is tailor-made for this case.

But if you insist on working from left to right, then complex analysis can help. The function $\sin (2k-1)x$ is secretly the imaginary part of $\exp((2k-1) i x)=z^{2k-1}$ where $z=\exp(ix)$. Then use the Taylor series

$$2\sum_{k=1}^\infty \frac{z^{2k-1}}{2k-1} = \log\frac{1+z}{1-z}$$

but watch out: our $z$ is right on the border of the disk of convergence. Finally, use the fact that the map $w= \frac{1+z}{1-z}$ sends the unit disk onto the right half-plane, and the imaginary part of $\log$ picks up the argument of $w$. Said argument is $\pm \pi/2$ when $w$ is on the boundary of the right half-plane...

The last, but not the least: try to type formulas instead of posting images. It's not that hard.