Question regarding complex power series

Question

Let us define a function $f$ on unit disc by-
$f(z)=\sum z^{2^n}\ \forall |z|<1$
Let $\theta=2\pi p/2^k$ where $p,k$ are positive integers and $0\le r<1$ so $f(re^{i\theta})$ exists.
Show that, $|f(re^{i\theta})|\to\infty$ as $r\to1$.
$f(re^{i\theta})=\sum_{n\ge0} r^{2^n} e^{i 2\pi p 2^{n-k}}=\sum_{n<k} r^{2^n} e^{i 2\pi p 2^{n-k}}$, since $e^{i 2\pi p 2^{n-k}}=1$ if $n\ge k$. Now I can't proceed further. Thanks for assistance in advance.

Answer 1

$f(re^{i\theta})=\sum_{n<k} r^{2^{n}} e^{i2 \pi p 2^{n-k}}+\sum_{n\geq k} r^{2^{n}} $ The first sum tends to a finite limit whereas the second sum tends to $\infty$.