# 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)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)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.