Alt. Series Test for Alt. Series of Real Numbers Examples 1

The Alternating Series Test for Alternating Series of Real Numbers Examples 1

Recall from The Alternating Series Test for Alternating Series of Real Numbers page the following test for convergence of an alternating series:

The Alternating Series Test

If $(a_n)_{n=1}^{\infty}$ is a decreasing sequence of real numbers and $\displaystyle{\lim_{n \to \infty} a_n = 0}$, then we have the following conclusion:

• The alternating series $\displaystyle{\sum_{n=1}^{\infty} (-1)^{n+1} a_n}$ converges.

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

Example 1

Show that $\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^n}{n + n^2}}$ converges.

We first notice that $(a_n)_{n=1}^{\infty} = \left ( \frac{1}{n + n^2} \right )_{n=1}^{\infty}$ is a decreasing sequence of positive real numbers. To show this, let $n \in \mathbb{N}$. Then $n^2 < (n+1)^2$ Furthermore, $n + n^2 < (n+1) + (n+1)^2$ and so:

(1)
\begin{align} \quad a_{n} = \frac{1}{n + n^2} > \frac{1}{(n+1) + (n+1)^2} = a_{n+1} \end{align}

Furthermore, it's not hard to see that:

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

So, by the alternating series test, $\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^n}{n + n^2}}$ converges.

Example 2

Show that $\displaystyle{\sum_{n=1}^{\infty} \frac{5\cos(n \pi)}{2^n + 3^n}}$ converges.

We note that the sequence $\displaystyle{(a_n)_{n=1}^{\infty} = \left ( \frac{5}{2^n + 3^n} \right )_{n=1}^{\infty}}$ is a decreasing sequence of real numbers. To show this, let $n \in \mathbb{N}$. Then $n < n+1$. So $2^n < 2^{n+1}$ and $3^n < 3^{n+1}$, so $2^n + 3^n < 2^{n+1} + 3^{n+1}$. Thus:

(3)
\begin{align} \quad a_n = \frac{5}{2^n + 3^n} > \frac{5}{2^{n+1} + 3^{n+1}} = a_{n+1} \end{align}

Furthermore we have that:

(4)
\begin{align} \quad \lim_{n \to \infty} \frac{5}{2^n + 3^n} = 0 \end{align}

Also notice that $\cos(n\pi) = (-1)^{n}$ for all $n \in \mathbb{N}$.

So, the alternating series $\displaystyle{\sum_{n=1}^{\infty} \frac{5\cos(n \pi)}{2^n + 3^n}}$ converges by the alternating series test.