Determine the radius of convergence 
of the power series:
![\[ \sum_{n=0}^{\infty} a^{n^2} z^n, \qquad 0 < a < 1. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-db852b206e73a6c419f0ca3188db9145_l3.png "Rendered by QuickLaTeX.com")
Test for convergence at the boundary points if is finite.
Let 
. Then,
![\begin{align*} \lim_{n \to \infty} b_n^{\frac{1}{n}} &= \lim_{n \to \infty} \left( \left( a^n z \right)^n \right)^{\frac{1}{n}} \\[9pt] &= \lim_{n \to \infty} a^n z \\[9pt] &= 0 \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-8f1385ec7baec67c75d0ac1edb987854_l3.png "Rendered by QuickLaTeX.com")
for all 
for any 
. Therefore, the radius of convergence is 
.
Determine the radius of convergence of the power series:
Test for convergence at the boundary points if is finite.
Let . Then,
for all for any . Therefore, the radius of convergence is .