The Root Test for Positive Series Examples 1

Recall from The Root Test for Positive Series of Real Numbers page the following test for convergence/divergence of a geometric series:

The Root Test for Positive Series of Real Numbers

Let $(a_n)_{n=1}^{\infty}$ be a positive sequence of real numbers and let $\displaystyle{L = \lim_{n \to \infty} (a_n)^{1/n}}$.

a) If $0 \leq L < 1$ then we conclude that:

• The series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.

b) If $1 < L \leq \infty$ then we conclude that:

• The series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.

c) If $L = 1$ then the ratio test is inconclusive.

We will now look at some examples of applying the root test.

Example 1

Determine whether $\displaystyle{\sum_{n=1}^{\infty} \left ( \frac{e^n}{n2^n} \right )^n}$ converges or diverges.

Applying the root test and we see that:

(1)

\begin{align} \quad L = \lim_{n \to \infty} \left ( \left ( \frac{e^n}{n2^n} \right )^n \right )^{1/n} = \lim_{n \to \infty} \frac{e^n}{n2^n} = \lim_{n \to \infty} \left ( \frac{e}{2} \right)^n \frac{1}{n} = \infty \end{align}

So by the root test we conclude that since $1 < L = \infty \leq \infty$ that $\displaystyle{\sum_{n=1}^{\infty} \left ( \frac{e^n}{n2^n} \right )^n}$ diverges.

Example 2

Determine whether $\displaystyle{\sum_{n=1}^{\infty} \frac{\sqrt{4^n}}{3^n}}$ converges or diverges.

Applying the root test and we see that:

(2)

\begin{align} \quad L = \lim_{n \to \infty} \left ( \frac{\sqrt{4^n}}{3^n} \right )^{1/n} = \lim_{n \to \infty} \frac{\sqrt[n+1]{4^n}}{3} = \frac{\sqrt{4}}{3} = \frac{2}{3} \end{align}

So by the root test we conclude that since $0 \leq \frac{2}{3} < 1$ that $\displaystyle{\sum_{n=1}^{\infty} \frac{\sqrt{4^n}}{3^n}}$ converges.