I have the following problem:
How smooth are the following functions? That is, how many derivatives can you guarantee them to have?
$$a)\;\;\;\;\; f(\theta)=\sum_{-\infty}^{\infty}\frac{e^{in\theta}}{n^{13.2}+2n^6-1}$$ $$b)\;\;\;\;\; f(\theta)=\sum_{0}^{\infty}\frac{\cos n\theta}{2^n}$$ $$c)\;\;\;\;\; f(\theta)=\sum_{0}^{\infty}\frac{\cos 2^n\theta}{2^n}$$
How should I approach this problem? Do I need to check whether the derivatives of the series are convergent or not perhaps?
The answers should be (from my book):
$$a)\;\;\;\;\;12$$ $$b)\;\;\;\;\;\infty$$ $$c)\;\;\;\;\;0$$