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)

\begin{align} \quad \lim_{n \to \infty} \frac{\frac{n + 1}{n!}}{\frac{n}{(n - 1)!}} = \lim_{n \to \infty} \frac{(n + 1)(n - 1)!}{n \cdot n!} = \lim_{n \to \infty} \frac{(n+1) \cdot 1 \cdot 2 \cdot ... \cdot (n - 1)}{n \cdot 1 \cdot 2 \cdot ... \cdot (n - 1) \cdot n } = \lim_{n \to \infty} \frac{n + 1}{n^2} = \lim_{n \to \infty} \frac{\frac{1}{n} + \frac{1}{n^2}}{1} = 0 \end{align}

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)

\begin{align} \quad \lim_{n \to \infty} \frac{\frac{(n+1)^2 (n+1)!}{(n + 2)!}}{\frac{n^2 n!}{(n + 1)!}} = \lim_{n \to \infty} \frac{(n + 1)^2 (n + 1)!(n + 1)!}{n^2 n! (n + 2)!} = \lim_{n \to \infty} \frac{(n + 1)^3 (n + 1)!}{n^2(n + 2)!} = \lim_{n \to \infty} \frac{(n + 1)^3}{n^2(n + 2)} = \lim_{n \to \infty} \frac{n^3 + 3n^2 + 3n + 1}{n^3 + 2n^2} = 1 \end{align}

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)

\begin{align} \quad \lim_{n \to \infty} \frac{\frac{5^{n+1}}{(n+1)!}}{\frac{5^n}{n!}} = \lim_{n \to \infty} \frac{5^{n+1} n!}{5^n (n+1)!} = \lim_{n \to \infty} \frac{5}{n + 1} = 0 \end{align}

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