Math 2.0

Functions of a complex variable Proposition

Radius of convergence of the binomial series

Builds on Binomial seriesRatio formula for the radius of convergence

Statement

Let α\alpha be a real number that is not a natural number. Then the binomial series ∑k=0∞(αk)xk\sum_{k=0}^{\infty} \binom{\alpha}{k} x^k has radius of convergence 11.

Remarks

The coefficients are nonzero, and (αk+1)/(αk)=(α−k)/(k+1)\binom{\alpha}{k+1} / \binom{\alpha}{k} = (\alpha - k)/(k + 1). Its absolute value tends to 11, so the ratio formula gives R=1R = 1. At x=±1x = \pm 1 the behavior depends on α\alpha. For α>0\alpha > 0 the series converges absolutely at both endpoints. Mathlib states the result as binomialSeries_radius_eq_one.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

Show the path as a graph
Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionThe sequence r^n tends to zero for |r| < 1, example, no evidence levelThe sequence r^n tend…exampleRadius of convergence, definition, no evidence levelRadius of convergencedefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionBinomial series, example, no evidence levelBinomial seriesexampleA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionSum of a series, definition, no evidence levelSum of a seriesdefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleRatio test, statement, no evidence levelRatio teststatementRatio formula for the radius of convergence, statement, no evidence levelRatio formula for the…statementRadius of convergence of the binomial series, statement, no evidence levelRadius of convergence…statement
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. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Geometric sequence definition
  5. Limit of a sequence definition
  6. Monotone sequence definition
  7. Series definition
  8. Sum of a finite geometric series statement
  9. Monotone convergence theorem for sequences statement
  10. Convergent sequence definition
  11. Squeeze theorem statement
  12. Partial sum definition
  13. Power series definition
  14. The sequence r^n tends to zero for |r| < 1 example
  15. Radius of convergence definition
  16. Convergent series definition
  17. Binomial series example
  18. A series of nonnegative terms converges iff its partial sums are bounded statement
  19. Absolute convergence definition
  20. Sum of a series definition
  21. Comparison test statement
  22. Geometric series example
  23. Ratio test statement
  24. Ratio formula for the radius of convergence statement
  25. Radius of convergence of the binomial series statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Binomial_series, Conditions for convergence
About this page
Tag0KX1MXJ5MR
URIm2://0KX1MXJ5MR
Kindstatement, proposition
WritingW0 stub
MSC30B10
StatusThis copy of the site has no build data. Run m2 status 0KX1MXJ5MR.
UsesBinomial series, Ratio formula for the radius of convergence

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.