# The Root Test for Positive Series Examples 2

If the sequence of terms $\{ a_n \}$ is ultimately positive and that $\lim_{n \to \infty} (a_n)^{1/n} = L$ where $L$ is some nonnegative real number or $\infty$, then recall from The Root Test for Positive Series page that if $0 ≤ L < 1$ then the series $\sum_{n=1}^{\infty} a_n$ converges. If $1 < L ≤ \infty$ then the series $\sum_{n=1}^{\infty} a_n$ diverges to infinity. If $L = 1$ then this test does not provide any additional information about the convergence/divergence of the series $\sum_{n=1}^{\infty} a_n$.

We are now going to look at some examples applying the root test.

## Example 1

**Using the root test, determine whether the series $\sum_{n=1}^{\infty} \frac{2^{n+1}}{n^n}$ converges or diverges.**

This series is clearly positive, and so we can apply the root test. We have that:

(1)Therefore the root test we have that $\sum_{n=1}^{\infty} \frac{2^{n+1}}{n^n}$ converges.

## Example 2

**Using the root test, determine whether the series $\sum_{n=1}^{\infty} \left ( \frac{n}{n + 1} \right )^{n^2}$ converges or diverges.**

Once again, we note that this is a positive series so we can apply the root test. We have that:

(2)Therefore by the root test we have that $\sum_{n=1}^{\infty} \left ( \frac{n}{n + 1} \right )^{n^2}$ converges.