11.5 Alternating Series

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11.5 Alternating Series

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Alternating Series

In this section we learn how to deal with series whose

terms are not necessarily positive. Of particular importance are alternating series, whose terms alternate in sign.

An alternating series is a series whose terms are

alternately positive and negative. Here are two examples:

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Alternating Series

We see from these examples that the nth term of an alternating series is of the form

a_n = (–1)^{n – 1}b_n or a_n = (–1)ⁿb_n where b_n is a positive number. (In fact, b_n= | a_n|.)

The following test says that if the terms of an alternating series decrease toward 0 in absolute value, then the series converges.

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Alternating Series

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Example 1

The alternating harmonic series

satisfies

(i) b_{n + 1} < b_nbecause (ii)

so the series is convergent by the Alternating Series Test.

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Estimating Sums

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Estimating Sums

A partial sum s_n of any convergent series can be used as an approximation to the total sum s, but this is not of much use unless we can estimate the accuracy of the

approximation. The error involved in using s ≈ s_n is the remainder R_n = s – s_n.

The next theorem says that for series that satisfy the conditions of the Alternating Series Test, the size of the error is smaller than b_{n + 1}, which is the absolute value of the first neglected term.

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Estimating Sums

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Example 4

Find the sum of the series correct to three decimal places.

Solution:

We first observe that the series is convergent by the Alternating Series Test because

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Example 4 – Solution

To get a feel for how many terms we need to use in our approximation, let’s write out the first few terms of the series:

cont’d

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Example 4 – Solution

Notice that

and

cont’d

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Example 4 – Solution

By the Alternating Series Estimation Theorem we know that

| s – s₆| ≤ b₇ < 0.0002

This error of less than 0.0002 does not affect the third decimal place, so we have s ≈ 0.368 correct to three decimal places.

cont’d

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Estimating Sums

Note:

The rule that the error (in using s_n to approximate s) is smaller than the first neglected term is, in general, valid only for alternating series that satisfy the conditions of the Alternating Series Estimation Theorem. The rule does not apply to other types of series.