Let $(a_n)$ be a sequence of real numbers such that $f(x) = \sum_{n=1}^{\infty} a_n x^{2^n}$ converges for $|x| < 1$ and $f(x)$ converges to $a$ as $x \to 1^{-}$. Then I have to prove that $\sum a_n$ exists and is equal to $a$. (This is a particular instance of so called High Indices Theorem, which was originally proved by Hardy and Littlewood.) Here, we are given a hint that, assuming $\sum a_n = a \ (A)$, we have $\sum a_n = a$ if and only if $\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} k a_k = 0.$ This is not hard to prove, by comparing the difference between the partial sum $s_N$ and $p_N = \sum a_n e^{-n/N}$.
The result follows once we show that $a_n \to 0$ as $n \to \infty$. But I failed to make any useful estimation except for a trivial bound $a_n \ll_{\varepsilon}(1+\varepsilon)^{2^n}$ obtained by considering the radius of convergence of $f$, and this is obviously of no use. I'm a bit depressed, because my professor told me it would follow from the hint above.
Can somebody give any idea or additional hint to me?