# The Ratio Test for Positive Series Examples 1

Suppose that $\{ a_n \}$ is an ultimately positive sequence and that $\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \rho$ where $\rho$ is a real number or $\rho$ is $\infty$. Recall from The Ratio Test for Positive Series page that:

- If $0 ≤ \rho < 1$ then the series $\sum_{n=1}^{\infty} a_n$ is convergent.

- If $1 < \rho ≤ \infty$ then the series $\sum_{n=1}^{\infty} a_n$ is divergent.

- If $\rho = 1$ then the ratio test fails as it provides no useful information regarding the convergence or divergence of a series. ||

We will now look at some examples applying the ratio test.

## Example 1

**Using the ratio test, determine whether the series $\sum_{n=1}^{\infty} \frac{n}{(n - 1)!}$ is convergent or divergent.**

Using the ratio test and noting that $a_n = \frac{n}{(n - 1)!}$ and $a_{n+1} = \frac{n + 1}{n!}$ we get that:

(1)So by the ratio test since $\rho = 0$, the series $\sum_{n=1}^{\infty} \frac{n}{(n - 1)!}$ is convergent.

## Example 2

**Using the ratio test, determine whether the series $\sum_{n=1}^{\infty} \frac{n^2 n!}{(n + 1)!}$ is convergent or divergent.**

We note that $a_n = \frac{n^2 n!}{(n + 1)!}$ and that $a_{n+1} = \frac{(n+1)^2 (n+1)!}{(n + 2)!}$. Applying the ratio test we get that:

(2)By the ratio test, since $\rho = 1$, we cannot determine whether this series is convergent or divergent.

## Example 3

**Using the ratio test, determine whether the series $\sum_{n=1}^{\infty} \frac{5^n}{n!}$ is convergent or divergent.**

We note that $a_n = \frac{5^n}{n!}$ and that $a_{n+1} = \frac{5^{n+1}}{(n+1)!}$. Applying the ratio test we get that:

(3)Therefore by the ratio test since $\rho = 0$, this series is convergent.