Nowhere-differentiable function via Fourier-Series

Question

Consider the function formally defined by $$f_\alpha(x) = \sum_{k=1}^\infty \frac{1}{k^\alpha}\sin (k^\alpha x).$$ For example, $f_1(x)$ is the $2\pi$-periodic extension of $$ x\mapsto \begin{cases}\hfill\frac{1}{2}(\pi-x),&\text{if }x\in[0,\pi]\\ \hfill -\frac12(\pi+x),&\text{if }x\in(-\pi,0)\end{cases}. $$ and obviously $\lim\limits_{\alpha\to\infty}f_\alpha(x) = \sin(x)$. It seems like e.g. $f_2(x)$ is an example of a nowhere differentiable function. So here's my question: How do I prove this and for what values of $\alpha$ is this true?