Math 2.0

Sequences, series, summability Definition

Alternating series

Builds on Series

Definition

Let (bn)(b_n) be a real sequence with bn>0b_n > 0 for all nn. The series ∑n=0∞(−1)nbn=b0−b1+b2−b3+⋯\sum_{n=0}^{\infty} (-1)^n b_n = b_0 - b_1 + b_2 - b_3 + \cdots is an alternating series. More generally, a series is alternating if its terms are nonzero and their signs alternate.

Remarks

Some texts allow bn≥0b_n \ge 0. The alternating harmonic series 1−1/2+1/3−⋯1 - 1/2 + 1/3 - \cdots is the standard example. The alternating series test gives a simple condition for convergence.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionAlternating series, definition, no evidence levelAlternating seriesdefinition
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  1. Sequence definition
  2. Real sequence definition
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7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Alternating_series, lead section
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