# The Radius of Convergence of a Power Series Examples 3

Recall from The Radius of Convergence of a Power Series page that we can calculate the radius of convergence of a power series $\sum_{n=0}^{\infty} a_n(x - c)^n$ using the ratio test, that is if $\lim_{n \to \infty} \biggr \rvert \frac{a_{n+1}}{a_n} \biggr \rvert = L$, then the radius of convergence is $R = \frac{1}{L}$. If $L = 0$ then the radius of convergence $R = \infty$ and if $L = \infty$ then the radius of convergence $R = 0$.

We will now look at some more examples of determining the radius of convergence of a given power series.

## Example 1

**Determine the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{3^n}{5^n \sqrt{n}} x^n$.**

We have that:

(1)Therefore the radius of convergence is $\frac{5}{3}$.

## Example 2

**Determine the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{2^n n! n^2 (3 + \sqrt{n})}{2 \cdot 3^{n+2} (n + 1)^2} (x - 2)^n$.**

We have that:

(2)The limit above equals $\infty$ because the degree in the numerator is greater than the degree in the denominator. Therefore the radius of convergence is $0$.