The Ratio Test for Positive Series of Real Numbers Examples 1

Recall from The Ratio Test for Positive Series of Real Numbers page the following test for convergence/divergence of a geometric series:

The Ratio Test for Positive Series of Real Numbers

Let $(a_n)_{n=1}^{\infty}$ be a positive sequence of real numbers and let $\displaystyle{\rho = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}}$.

a) If $0 \leq \rho < 1$ then we conclude that:

• The series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.

b) If $1 < \rho \leq \infty$ then we conclude that:

• The series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.

c) If $\rho = 1$ then the ratio test is inconclusive.

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

Example 1

Determine whether $\displaystyle{\sum_{n=1}^{\infty} \frac{e^nn!}{3n2^n(n+1)!}}$ converges or diverges.

Using the ratio test and we see that:

(1)

\begin{align} \quad \rho = \lim_{n \to \infty} \left ( \frac{e^{n+1}(n+1)!}{3(n+1)2^{n+1}(n+2)!} \cdot \frac{3n2^n(n+1)!}{e^nn!}\right ) = \lim_{n \to \infty} \frac{en(n+1)}{2(n+1)(n+2)} = \frac{e}{2} \end{align}

So $\displaystyle{\rho = \frac{e}{2} > 1}$. Hence $\displaystyle{\sum_{n=1}^{\infty} \frac{e^nn!}{3n2^n(n+1)!}}$ diverges by the ratio test.

Example 2

Determine whether $\displaystyle{\sum_{n=1}^{\infty} \frac{2^{n^2}cos \left ( \frac{1}{n} \right )}{e^n(n+1)!}}$ converges or diverges.

Using the ratio test and we see that:

(2)

\begin{align} \quad \rho = \lim_{n \to \infty} \left ( \frac{2^{(n+1)^2}cos \left ( \frac{1}{(n+1)} \right )}{e^{n+1}(n+2)!} \cdot \frac{e^n(n+1)!}{2^{n^2}cos \left ( \frac{1}{n} \right )} \right ) = \lim_{n \to \infty} \frac{2^{2n+1} \cos \left ( \frac{1}{n+1} \right )}{e(n+2)\cos \left ( \frac{1}{n} \right )} = \lim_{n \to \infty} \frac{2^{2n+1}}{e(n+2)} = \infty \end{align}

so $\displaystyle{\rho = \infty > 0}$. Hence $\displaystyle{\sum_{n=1}^{\infty} \frac{2^{n^2}cos \left ( \frac{1}{n} \right )}{e^n(n+1)!}}$ diverges by the ratio test.