1. https://www. khanacademy

2. 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.

3. Verify that the series

5. 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.

6. 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.

7. 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

9. 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

11. 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.

12. 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

13. 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