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In the previous section, we studied positive series, but we still lack the tools to analyze series with both positive and negative terms. One of the keys to understanding such series is the concept of absolute convergence.

Verify that the series

The series in the previous example does not converge absolutely, but we still do not know whether or not it converges. A series may converge without converging absolutely. In this case, we say that is conditionally convergent. DEFINITION Conditional Convergence An infinite series converges conditionally if converges but diverges.

Alternating Series where the terms an are positive and decrease to zero. An alternating series with terms decreasing in magnitude. The sum is the signed area, which is less than a1.

Then the following alternating series converges: THEOREM 2 Leibniz Test for Alternating Series Assume that {an} is a positive sequence that is decreasing and converges to 0: Then the following alternating series converges: Furthermore, Example Next Example

converges conditionally and that 0 < S < 1. Show that converges conditionally and that 0 < S < 1. The terms are positive and decreasing, and Therefore, S converges by the Leibniz Test. Furthermore, 0 < S < 1. However, the positive series diverges. Therefore, S is conditionally convergent but not absolutely convergent. (A) Partial sums of S = (B) Partial sums

Alternating Harmonic Series Show that converges conditionally. Then: (a) Show that (b) Find an N such that SN approximates S with an error less than 10−3.

Alternating Harmonic Series Show that converges conditionally. Then: (a) Show that (b) Find an N such that SN approximates S with an error less than 10−3. We can make the error less than 10−3 by choosing N so that

CONCEPTUAL INSIGHT The convergence of an infinite series depends on two factors: (1) how quickly an tends to zero, and (2) how much cancellation takes place among the terms. Consider