AP Calculus BC Review Alternating Series

An alternating series is one in which the signs of the terms switch between positive and negative. These kinds of series show up fairly regularly in applications. So its important to know how to work with them. In this review article, well examine the properties of alternating series. Well also work through a number of examples similar to those you might find on the AP Calculus BC exam. Sign, Sign, Everywhere a Sign What makes a series alternating is the pattern of its signs. Positive terms alternate with negative terms forever. The first term may be either positive or negative. So, if b1, b2, b3, b3, etc., are positive, then both of the following are alternating series. For example, the following series is alternating. Heres another example. Notice again how the factor of (-1)n switches the sign. Here is one that is not easy to tell is alternating. In fact, its not obvious until you work out the value of each term. Remember your unit circle! Alternating Series Test There is actually a very simple test for convergence that applies to many of the series that youll encounter in practice. Suppose that an is an alternating series, and let bn = |an|. Then the series converges if both of the following conditions hold. The sequence of (positive) terms bn eventually decreases. That means that perhaps ignoring a few stray terms at the beginning, we have bn bn+1 bn+2 bn+3 The limit of the sequence (bn) is equal to zero. That is, Basically, if a series alternates, then as long as the terms get closer to zero, there must be a finite sum. Think of a bouncing ball. Each up bounce is a positive term, and each downward return is a negative term. If the next bounce is always smaller than the previous one, then eventually the ball will come to rest. Bouncing Ball (by MichaelMaggs, edit by Richard Bartz, via Wikimedia Commons) Error Bound Directly related to the convergence test, there is an easy error estimate for these kinds of series. If an is a convergent alternating series, then the nth partial sum, sn, approximates the sum of the series to within an error bound of |an+1|. Note that the error bound does not apply to divergent series. Example The Alternating Harmonic Series The alternating harmonic series is the alternating sum of the reciprocals of all the natural numbers. That is, Does this series converge? Lets use the Alternating Series Test to find out. Here, bn = |an| = 1/n, which decreases to 0 as n . Thus, the alternating harmonic sequence converges. (on the other hand, the plain old harmonic series, which consists of all positive fractions, actually diverges!) How close is 1 1/2 + 1/3 1/, and I agree! Adding and subtracting almost two hundred fractions is not a very efficient approach for estimating to within 0.01 accuracy. (Fortunately, there are better ways to find the digits of in practice.) However, this particular example does highlight the theoretic importance of alternating series and their properties. Summary An alternating series is a series in which the signs of the terms alternate between positive and negative forever. The Alternating Series Test states that such a series will converge if the sequence of the absolute values of its terms decreases to zero in the limit. The error for the nth partial sum is bounded by |an+1|.