Is the function given by $\displaystyle \sum\limits_{n=1}^{\infty} (-1)^{n} \frac{\cos{nx}}{n^{3/2}}$ continuous for $\displaystyle x \in [-\pi, \pi]$.
Is the function given by $\displaystyle \sum\limits_{n=1}^{\infty} (-1)^{n} \frac{\cos{nx}}{n^{3/2}}$ continuous for $\displaystyle x \in [-\pi, \pi]$.
Since the series converges even absolutely for all x, this series defines a function that's even derivable everywhere.