Home » Blog » Determine the radius of convergence of ∑ (3n1/2 zn / n)

Determine the radius of convergence of ∑ (3n1/2 zn / n)

Determine the radius of convergence r of the power series:

    \[ \sum_{n=1}^{\infty} \frac{3^{\sqrt{n}} z^n}{n}. \]

Test for convergence at the boundary points if r is finite.


Let a_n = \frac{3^{\sqrt{n}} z^n}{n}. 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*}

Thus, the radius of convergence is r = 1, and the series converges for z such that |z| < 1.

One comment

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):