Alternating series are series whose terms alternate in sign between positive andnegative. There is a powerful convergence test for alternating series.
Many of the series convergence tests that have been introduced so far are stated withthe assumption that all terms in the series are nonnegative. Indeed, this condition isassumed in the Integral Test, Ratio Test, Root Test, Comparison Test and LimitComparison Test. In this section, we study series whose terms are not assumed to bestrictly positive. In particular, we are interested in series whose terms alternatebetween positive and negative (aptly named alternating series). It turnsout that there is a powerful test for determining that a series of this formconverges.
Let be a sequence of positive numbers. An alternating series is a series of the form or of the form
As usual, this definition can be modified to include series whose indexing startssomewhere other than .
The geometric series is alternating. In general, a geometric series with ratio is alternating, since
Recall the harmonic series which is perhaps the simplest example of a divergent series whose terms approachzero as approaches . A similarly important example is the alternating harmonicseries The terms of this series, of course, still approach zero, and their absolute values aremonotone decreasing. Because the series is alternating, it turns out that this isenough to guarantee that it converges. This is formalized in the followingtheorem.
Alternating Series Test Let be a sequence whose terms are eventually positive andnonincreasing and . Then, the series both converge.
Compared to our convergence tests for series with strictly positive terms, this test isstrikingly simple. Let us examine why it might be true by considering the partialsums of the alternating harmonic series. The first few partial sums with odd indexare given by
Note that a general odd partial sum is of the form and the quantity in the parentheses is positive. We conclude that:
Moreover, the sequence of odd partial sums is bounded below by zero, since and each quantity in parentheses is positive.
We conclude that the sequence of odd partial must converge to a finite limit byapplying
The Fundamental Theorem of Calculus. The Monotone ConvergenceTheorem. The Ratio Test.
Next consider the sequence
of even partial sums.
The sequence of even partial sums defined above is
increasing and bounded, andtherefore converges by the Monotone Convergence Theorem. decreasing andbounded, and therefore converges by the Monotone Convergence Theorem.
Finally, we use and the fact that the limits of the sequences and are finite to concludethat It follows that the alternating harmonic series converges.
With slight modification, the argument given above can be used to prove that theAlternating Series Test holds in general.
The terms of the sequence are positive and nonincreasing, so we can apply theAlternating Series Test. Since the Alternating Series Test implies that the series converges.
Does the alternating series test apply to the series
yes no
The underlying sequence is . This sequence is positive and approaches as . However,it is not a decreasing sequence; the value of oscillates between and as . We cannotremove a finite number of terms to make decreasing, therefore we cannot apply thealternating series test.
Keep in mind that this does not mean we conclude the series diverges; in fact, it doesconverge. We are just unable to conclude this based on the alternating series test.
