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)

\begin{align} \quad \lim_{n \to \infty} \left ( \frac{2^{n+1}}{n^n} \right )^{1/n} = \lim_{n \to \infty} \frac{2^{\frac{n+1}{n}}}{n} = \lim_{n \to \infty} \frac{2^{1 + \frac{1}{n}}}{n} = 0 \end{align}

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)

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

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