Math 2.0

Sequences, series, summability Special summability methods Theorem

Abel's theorem

Builds on Power seriesConvergent seriesOne-sided limit

Statement

Let (an)(a_n) be a real sequence such that the series ∑n=0∞an\sum_{n=0}^{\infty} a_n converges. Let G(x)=∑n=0∞anxnG(x) = \sum_{n=0}^{\infty} a_n x^n for −1<x<1-1 < x < 1. Then G(x)→∑n=0∞anG(x) \to \sum_{n=0}^{\infty} a_n as x→1x \to 1 from the left.

Remarks

The power series converges on (−1,1)(-1, 1), since its radius of convergence is at least 11. The theorem gives the sum of the alternating harmonic series. Apply it to the Mercator series for ln⁡(1+x)\ln(1 + x). The converse fails. For an=(−1)na_n = (-1)^n, G(x)=1/(1+x)→1/2G(x) = 1/(1 + x) \to 1/2, but ∑(−1)n\sum (-1)^n diverges. Mathlib states it as Real.tendsto_tsum_powerSeries_nhdsWithin_lt.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbel's theorem, statement, no evidence levelAbel's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. One-sided limit definition
  6. Limit of a sequence definition
  7. Series definition
  8. Convergent sequence definition
  9. Partial sum definition
  10. Power series definition
  11. Convergent series definition
  12. Abel's theorem statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Abel%27s_theorem, Theorem
About this page
Tag0S4EMDQS2F
URIm2://0S4EMDQS2F
Aliasesabel-limit-theorem
Kindstatement, theorem
WritingW1 stated
MSC40A30
StatusThis copy of the site has no build data. Run m2 status 0S4EMDQS2F.
UsesPower series, Convergent series, One-sided limit
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