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

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.

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Ex.

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

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

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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

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.

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

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