Lesson 16 - Alternating Series and Alternating Series Test

Alternating Series

A series whose terms alternate between positive and negative values is an alternating series. For example, the series

\sum_{n = 1}^{\infty} {(- \frac{1}{2})}^{n} = - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{6} - \dots

and

\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots

are both alternating series.

Any series whose terms alternate between positive and negative values is called an alternating series. An alternating series can be written in the form

\sum_{n = 1}^{\infty} (- 1)^{n + 1} b_{n} = b_{1} - b_{2} + b_{3} - b_{4} + \dots

\sum_{n = 1}^{\infty} (- 1)^{n} b_{n} = - b_{1} + b_{2} - b_{3} + b_{4} - \dots

Where $b_{n} > 0$ for all positive integers $n$ .

See a proof here.

Alternating Series Test

An alternative series of the form

\sum_{n = 1}^{\infty} (- 1)^{n + 1} b_{n} o r \sum_{n = 1}^{\infty} (- 1)^{n} b_{n}

converges if
i. $0 < b_{n + 1} \leq b_{n}$ for all $n \geq 1$ and
ii. $lim_{n \to \infty} b_{n} = 0$ .

This is known as the alternating series test.

Example: Convergence of Alternating Series

For each of the following alternating series, determine whether the series converges or diverges.

a. $\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{n^{2}}$
b. $\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1} n}{n + 1}$

a. Since
$\frac{1}{(n + 1)^{2}} < \frac{1}{n^{2}}$ and $\frac{1}{n^{2}} \to 0$
the series converges.
b. Try for yourself.

Remainder of an Alternating Series

Sometimes, it is difficult to explicitly calculate the sum of an alternating series. Instead, we generally use an approximated sum, called the partial sum. Remainder estimates give us a way to control the error in our approximation of the sum.

Consider an alternating series in the form of

\sum_{n = 1}^{\infty} (- 1)^{n + 1} b_{n} o r \sum_{n = 1}^{\infty} (- 1)^{n} b_{n}

that satisfies the hypotheses of the alternating series test. Let S denote the sum of the series and $S_{N}$ denote the $N t h$ partial sum. For any integer $N \geq 1$ , the remainder $R_{N} = S - S_{N}$ satisfies

| R_{N} | \leq b_{N + 1} .

Example: Estimating the Remainder of an Alternating Series

Consider the alternating series

\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{n^{2}}

Use the remainder estimate to determine a bound on the error $R_{10}$ if we approximate the sum of the series by the partial sum $S_{10}$ .

Solution

From the theorem stated above,
$| R_{10} | < b_{11} = \frac{1}{11^{2}} \approx 0.008265$ .

Absolute and Conditional Convergence

Consider a series $\sum_{n = 1}^{\infty} a_{n}$ and the related series $\sum_{n = 1}^{\infty} | a_{n} |$ .

A series $\sum_{n = 1}^{\infty} a_{n}$ is exhibits absolute convergence if $\sum_{n = 1}^{\infty} | a_{n} |$ converges. A series $\sum_{n = 1}^{\infty} a_{n}$ exhibits conditional convergence if $\sum_{n = 1}^{\infty} a_{n}$ but $\sum_{n = 1}^{\infty} | a_{n} |$ diverges.

Theorem: Absolute Convergence Implies Convergence

If $\sum_{n = 1}^{\infty} | a_{n} |$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges.

Here is a proof for absolute convergence.

Example: Absolute versus Conditional Convergence

For each of the following series, determine whether the series converges absolutely, converges conditionally, or diverges.

a. $\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{(3 n + 1)}$
b. $\sum_{n = 1}^{\infty} \frac{\cos (n)}{n^{2}}$

a. We can see that

\sum_{n = 1}^{\infty} | \frac{(- 1)^{n + 1}}{3 n + 1} | = \sum_{n = 1}^{\infty} \frac{1}{3 n + 1}

diverges by using the limit comparison test with the harmonic series.

lim_{n \to \infty} \frac{\frac{1}{(3 n + 1)}}{\frac{1}{n}} = \frac{1}{3}

Applying the theorem, the series cannot converge absolutely. Moreover, because of the alternating series test, we can see that the series converges.

\frac{1}{3 (n + 1) + 1} < \frac{1}{3 n + 1} a n d \frac{1}{3 n + 1} \to 0

We can conclude that, since the alternating series test confirms convergence but the absolute series diverges, $\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{(3 n + 1)}$ converges conditionally.

b. Try for yourself.

Next, we will review Power Series.

Next Lesson: Lesson 17 - Power Series

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Note: This lesson adapted from OpenStax. Credit goes to OpenStax creators.