Determine the radius of convergence 
of the power series:
![\[ \sum_{n=1}^{\infty} \frac{3^{\sqrt{n}} z^n}{n}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-74a71dcf96f7fe0fdecd215a9b09fa6d_l3.png "Rendered by QuickLaTeX.com")
Test for convergence at the boundary points if is finite.
Let 
. Then,
![\begin{align*} \lim_{n \to \infty} a_n^{\frac{1}{n}} &= \lim_{n \to \infty} \left( \frac{3^{\sqrt{n}} z^n}{n} \right)^{\frac{1}{n}} \\[9pt] &= \lim_{n \to \infty} \frac{3^{\frac{1}{\sqrt{n}}} z}{n^{\frac{1}{n}}} \\[9pt] &= z \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-f54e37ff0fb2996d5e36774bb0e84a45_l3.png "Rendered by QuickLaTeX.com")
Thus, the radius of convergence is 
, and the series converges for 
such that 
.
Determine the radius of convergence of the power series:
Test for convergence at the boundary points if is finite.
Let . Then,
Thus, the radius of convergence is , and the series converges for such that .
It diverges in the boundary points in comparison to 1/n