The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Alternating Series Test


Assume that

(i) 09_infinite_series-278.gif is an alternating series.

(ii) The terms an are decreasing, a1 > a2 > ... > an > ....

(iii) The terms approach zero, limn→∞ an = 0.

Then the series converges to a sum

09_infinite_series-279.gif = S.

Moreover, the sum S is between any two consecutive partial sums,

S2n< S < S 2n+1.


We see from the graph in Figure 9.5.1 that the partial sums Sn alternately increase and decrease, but the change is less each time. The value of Sn "vibrates" back and forth and the vibration damps down around the limit S.


The sequence of even partial sums is increasing.

S2 < 54 < ... < S2n < ...,


S4 = S2 + (a3 - a4), S6 = S4 + (a5 - a6), etc.

The sequence of odd partial sums is decreasing,

S1 > S3 > S5 > ...,


S3 = S1 - (a2 - a3), S5 = S3 - (a4 - a5), etc.


Figure 9.5.1

It follows that each even partial sum is less than Sx,

S1 > S1 - a2 = S2, S1 > S3 - a4 = S4, S1 > S5 - a6 = S6, etc.

Theorem 1 (Section 9.4) shows that the increasing sequence of even partial sums converges,

limn→∞ S2n = S.

Given any infinite H, a2H+l ≈ 0 and S2H ≈ S, so

S2H+1 = S2H + a2H+1 ≈ S

Therefore the sequence of all partial sums converges to S, and


Finally, since the even partial sums are increasing and the odd partial sums are decreasing, we have the estimate

S2n < S < S2n+1.

Figure 9.5.2 shows a graph of the partial sums.


Figure 9.5.2

Example 1: Alternating Harmornic Series Converging
Example 2: Diverging Alternating Series

The Cauchy Test for Divergence in Section 9.2 shows that if the terms an do not converge to zero the series diverges.

We have now built up quite a long list of convergence tests. The next section contains one more important test, the Ratio Test. At the end of that section is a summary of all the convergence tests with hints on when to use them.

Last Update: 2006-11-07