The Radius of Convergence of a Power Series Examples 2

Recall from The Radius of Convergence of a Power Series page that we can calculate the radius of convergence of a power series 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 $\sum_{n=1}^{\infty} \frac{x^n}{4^n \ln n}$.

Applying the ratio test and we get that:

(1)

\begin{align} \quad \lim_{n \to \infty} \biggr \rvert \frac{a_{n+1}}{a_n} \biggr \rvert = \lim_{n \to \infty} \biggr \rvert \frac{4^n \ln (n) }{4^{n+1} \ln (n + 1)} \biggr \rvert = \lim_{n \to \infty} \frac{\ln (n)}{4 \ln (n + 1)} \end{align}

Let $y = \frac{\ln (x)}{4 \ln (x + 1)}$. Then applying L'Hospital's rule we get that:

(2)

\begin{align} \quad \lim_{x \to \infty} \frac{\ln (x)}{4 \ln (x + 1)} \overset H = \lim_{x \to \infty} \frac{\frac{1}{x}}{4 \frac{1}{x +1}} = \lim_{x \to \infty} \frac{x + 1}{4x} = \frac{1}{4} \end{align}

Therefore the radius of convergence is $4$.

Example 2

Determine the radius of convergence of the power series $\sum_{n=1}^{\infty} \frac{(3x - 2)^n}{n3^n}$.

We must first rewrite the series above as follows:

(3)

\begin{align} \quad \sum_{n=1}^{\infty} \frac{(3x - 2)^n}{n3^n} = \sum_{n=1}^{\infty} \frac{\left (3 \left (x - \frac{2}{3} \right ) \right )^n}{n3^n} = \sum_{n=1}^{\infty} \frac{3^n \left (x - \frac{2}{3} \right )^n}{n3^n} = \sum_{n=1}^{\infty} \frac{\left (x - \frac{2}{3} \right )^n}{n} \end{align}

Now applying the ratio test and we get that:

(4)

\begin{align} \quad \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{n}{n+1} = 1 \end{align}

Therefore the radius of convergence is $1$.

Example 3

Determine the radii of convergence of the power series $\sum_{n=1}^{\infty} \frac{(ax + b)^n}{c^n}$. where $a, b, c \in \mathbb{R}$ and $a, c \neq 0$.**

We will first rewrite the series above as follows:

(5)

\begin{align} \quad \sum_{n=1}^{\infty} \frac{(ax + b)^n}{c^n} = \sum_{n=1}^{\infty} \frac{\left (a \left (x + \frac{b}{a} \right ) \right )^n}{c^n} = \frac{a^n \left ( x + \frac{b}{a} \right )^n}{c^n} \end{align}

Therefore applying the ratio test we get that:

(6)

\begin{align} \quad \lim_{n \to \infty} \biggr \rvert \frac{a^{n+1}c^n}{a^{n}c^{n+1}} \biggr \rvert = \lim_{n \to \infty} \biggr \rvert \frac{a}{c} \biggr \rvert \end{align}

Therefore the radius of convergence is $\biggr \rvert \frac{c}{a} \biggr \rvert$.