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Sequences & Series · Convergence Tests

Alternating Series Test

∑(−1)nbn converges if bn>0,  bn+1≤bn,  lim⁡bn=0\sum (-1)^n b_n \text{ converges if } b_n > 0,\; b_{n+1} \leq b_n,\; \lim b_n = 0

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}|.

Worked examples

Does Σ (-1)ⁿ⁺¹/n converge?
  1. bₙ = 1/n > 0 ✓
  2. 1/(n+1) ≤ 1/n (decreasing) ✓
  3. lim 1/n = 0 ✓

Answer: Converges (this is the alternating harmonic series, sum = ln 2).

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