The Second Slow-Converging Case

Consider the odd function defined by the very slowly convergent sine series

$$S_{\rm s} = \frac{2}{\pi} \sum_{k=1}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\, \sin(k\theta).$$

The corresponding FC series is then

$$S_{\rm c} = \frac{2}{\pi} \sum_{k=1}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\, \cos(k\theta),$$

and the complex power series $S_{z}$ is given by

$$S_{z} = \frac{2}{\pi} \sum_{k=1}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\, z^{k},$$

of which the two DP Fourier series above are the real and imaginary parts on the unit circle.