Region of convergence for the power series $\sum_{n=1}^\infty x^{(n^2)}$

Question

I am trying to find the region of convergence for the (real) power series $\sum_{n=1}^\infty x^{(n^2)}$. My guess is that it is $x\in (-1,1)$ but I haven't been able to prove this. The only theorems I know (and have proved) in this context are Cauchy–Hadamard and d'Alembert's and neither seem applicable.

Hints, rather then detailed solutions, would be greatly appreciated.

Answer 1

Instead of trying to apply big theorems here it's easier to just figure out when the series converges. Hints: (i) If $|x|<1$ then $\left|x^{n^2}\right|\le|x|^n$, (ii) If $|x|\ge1$ then $\lim_{n\to\infty}x^{n^2}=\dots$.

Answer 2

By Root Test we have $\lim_{n\rightarrow\infty}\left(|x|^{n^{2}}\right)^{1/n}=\lim_{n\rightarrow\infty}|x|^{n}=0$ for $|x|<1$ and $=\infty$ for $|x|>1$.