Math 2.0

Functions of a complex variable Series expansions Theorem

Cauchy–Hadamard theorem

Builds on Radius of convergenceLimit superiorRoot test

Statement

Let ∑an(x−c)n\sum a_n (x - c)^n be a real power series with radius of convergence RR. Then 1R=lim sup⁡n→∞∣an∣1/n,\frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n}, with the conventions 1/0=∞1/0 = \infty and 1/∞=01/\infty = 0.

Remarks

The proof applies the root test to the series ∑an(x−c)n\sum a_n (x - c)^n. The limit superior of ∣an(x−c)n∣1/n|a_n (x - c)^n|^{1/n} is ∣x−c∣lim sup⁡∣an∣1/n|x - c| \limsup |a_n|^{1/n}. For example, ∑nxn\sum n x^n has R=1R = 1, since n1/n→1n^{1/n} \to 1. Mathlib states it as FormalMultilinearSeries.radius_inv_eq_limsup.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

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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 levelSeriesdefinitionLimit superior, definition, no evidence levelLimit superiordefinitionSum 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 seriesdefinitionA 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 seriesexampleRoot test, statement, no evidence levelRoot teststatementCauchy–Hadamard theorem, statement, no evidence levelCauchy–Hadamard theor…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. Limit superior definition
  9. Sum of a finite geometric series statement
  10. Monotone convergence theorem for sequences statement
  11. Convergent sequence definition
  12. Squeeze theorem statement
  13. Partial sum definition
  14. Power series definition
  15. The sequence r^n tends to zero for |r| < 1 example
  16. Radius of convergence definition
  17. Convergent series definition
  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. Root test statement
  24. Cauchy–Hadamard theorem statement

Where this leads

28 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy–Hadamard_theorem, Theorem for one complex variable
About this page
Tag07E5MB1CA2
URIm2://07E5MB1CA2
Aliasescauchy-hadamard-formula
Kindstatement, theorem
WritingW1 stated
MSC30B10
StatusThis copy of the site has no build data. Run m2 status 07E5MB1CA2.
UsesRadius of convergence, Limit superior, Root test
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