Sequences & Series · Convergence Tests
Alternating Series Test
An alternating series converges if the absolute values of terms are decreasing and approach zero. The error after N terms is at most |b_{N+1}|.
- 2 worked examples
- 18 in Sequences & Series
Worked examples
- bₙ = 1/n > 0 ✓
- 1/(n+1) ≤ 1/n (decreasing) ✓
- lim 1/n = 0 ✓
Answer: Converges (this is the alternating harmonic series, sum = ln 2).
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Common questions
What is the Alternating Series Test?
An alternating series converges if the absolute values of terms are decreasing and approach zero. The error after N terms is at most |b_{N+1}|. It is one of the sequences & series formulas in the CalcRef reference.
How do you use the Alternating Series Test?
Worked example. Does Σ (-1)ⁿ⁺¹/n converge? bₙ = 1/n > 0 ✓. 1/(n+1) ≤ 1/n (decreasing) ✓. lim 1/n = 0 ✓. Answer: Converges (this is the alternating harmonic series, sum = ln 2).
Practice
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