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> I'm reading the problems of Stein and Shakarchi's Complex Analysis, Chapter 2 Problem 1 asks to show that $$f(z)=\sum\_{n=0}^{\infty}z^{2^n}$$ cannot be analytically continued past the unit disk. (Hint: Suppose $\theta =\frac{2\pi p}{2^k}$ for $p,k$ positive integers, let $z=re^{i\theta}$ and show $\mid f(z)\mid\rightarrow\infty$ as $r \rightarrow1$).
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I understand from the hint they want me to "pepper" the unit circle with points where the power expansion explodes so that it is dense with poles. I do not understand why they choose such particular points, but I assume that in retrospect it will show that those are the ones that I can show divergence for the easiest and are dense in the unit circle, so plowing ahead:

$$\lim\_{r \rightarrow 1}\left| \sum\_{n=0}^\infty r^{2^n}e^{ i2\pi p 2^{n-k}}\right| = \left| \sum\_{n=0}^k e^{ \frac{i2\pi p}{2^{k-n}}} + \sum\_{n=k+1}^\infty e^{ i2\pi p2^{n-k}} \right| $$

Where do I go from here? Is there some oversimplification of these sinusoids that I'm not seeing? Furthermore, once I manage to show this explodes, if I show that these numbers are dense on the unit circle I'm done, right?

Any insight is much appreciated.