1549901369-Elements_of_Real_Analysis__Denlinger_

478 Chapter 8 • Infinite Series of Real Numbers

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Therefore, {Sn} converges (see Exercise 2.6.7). That is, .2,)-l)n+lan con-
n=l
verges. •

Example 8.3.3 The alternating harmonic series

co nverges, by the alternating series test.

In the case of an alternating series, we can do more than tell whether it
converges. If it converges, we can calculate the actual sum of the series to any
specified degree of accuracy. The following theorem tells how this can be done.

Theorem 8.3.4 (Sum of Alternating S eries) If {an} is a monotone de-

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creasing sequence of positive numbers with an ----; 0, and S = 2:)-l)n+lan,

n=l

then, \:/n EN,

(a) S2n < S < S2n+i, and

Proof. Suppose {an} is a monotone decreasing sequence of positive num-

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bers with an ----; 0. By Theorem 8.3.2, S = .2,)-1t+^1 an exists. Let n E N.

n=l

Then,

S = S2n + a2n+1 - a2n+2 + a2n+3 - a2n+4 +

> S2n· (8)

From these relations we also have

(9)

Similarly,

S = S2n+1 - (a2n+2 - a2n+3) - (a2n+4 - a2n+s) -

(10)