# Prove Holder-continuity for $\mu_\lambda (x) = \sum\limits_{n=1}^\infty \frac{ \cos(2^n x)}{2^{n \lambda} }$

Consider $\lambda \in (0,1]$ and the function $\mu_\lambda$ defined by $\mu_\lambda (x) = \sum\limits_{n=1}^\infty \frac{ \cos(2^n x)}{2^{n \lambda} }$ for $x \in [0,\pi]$.  Show that  $\mu_\lambda$  is  $\alpha$-Holder-continuous on $[0, \pi]$ for all  $0 < \alpha < \lambda$.

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