1. Alternating Series
An alternating series is a series where terms alternate in sign.

2. Thm. Alternating Series Test for Convergence
The alternating series Ʃ(-1)nan will converge if i)
ii) an is a decreasing sequence
(an + 1 < an after some value of n)
 Be sure to verify conditions before using this test.

3. Ex.

4. Ex.

5. Thm. Absolute Convergence Test
If Ʃ|an| converges, then Ʃan converges.
Ex.

6. Ʃan is absolutely convergent if Ʃ|an| and Ʃan both converge.
Ʃan is conditionally convergent if Ʃan converges but Ʃ|an| diverges.

7. Ex.

8. Ex.

9. Checking Convergence of Σ(-1)nan
Determine convergence of Σan
Σan convergent
Σan divergent
Σ(-1)nan abs. conv. by Abs. Conv. Test
Check for cond. conv. of Σ(-1)nan using Alt. Series Test

10. Thm. Alternating Series Remainder
If you use N terms to approximate the sum of the convergent alternating series Σ(-1)nan, then the error is less than aN + 1.

11. Ex. Approximate the value of
using the first 6 terms. How many terms are needed to ensure an error less than .001?

12. 3. Find with an error less than .001
Pract.
Determine if the series is absolutely convergent, conditionally convergent, or divergent. State the test used. 1. 2.
3. Find with an error less than .001
Cond. Conv.; p-series, LCT, and AST
Abs. Conv.; Geom., LCT, and ACT
0.368