Math 2.0

Sequences, series, summability Series: basic notions Example

Alternating harmonic series

Builds on Alternating series testHarmonic series

Example

The alternating harmonic series is ∑n=1∞(−1)n+1n=1−12+13−14+⋯ .\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots. It converges by the alternating series test, since 1/n1/n decreases to 00. Its sum is ln⁡2≈0.6931\ln 2 \approx 0.6931.

Remarks

The series of absolute values is the harmonic series, which diverges. So the convergence is conditional. The value ln⁡2\ln 2 comes from the Mercator series ln⁡(1+x)=x−x2/2+x3/3−⋯\ln(1 + x) = x - x^2/2 + x^3/3 - \cdots at x=1x = 1, with Abel's theorem. By the Riemann rearrangement theorem, a reordering of the terms can give any real sum.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementAlternating series, definition, no evidence levelAlternating seriesdefinitionPartial sum, definition, no evidence levelPartial sumdefinitionAlternating series test, statement, no evidence levelAlternating series te…statementConvergent series, definition, no evidence levelConvergent seriesdefinitionDivergent series, definition, no evidence levelDivergent seriesdefinitionHarmonic series, example, no evidence levelHarmonic seriesexampleAlternating harmonic series, example, no evidence levelAlternating harmonic …example
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  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Limit of a sequence definition
  5. Monotone sequence definition
  6. Series definition
  7. Convergent sequence definition
  8. Monotone convergence theorem for sequences statement
  9. Alternating series definition
  10. Partial sum definition
  11. Alternating series test statement
  12. Convergent series definition
  13. Divergent series definition
  14. Harmonic series example
  15. Alternating harmonic series example

Further reading

  1. Web https://en.wikipedia.org/wiki/Natural_logarithm_of_2, Series representations
  2. Web https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), Alternating harmonic series
About this page
Tag0JQ5Y3WWVE
URIm2://0JQ5Y3WWVE
Kindexample
WritingW2 usable
MSC40A05
Instance ofConvergent series, Alternating series
UsesAlternating series test, Harmonic series

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