Math 2.0

statement proposition

Ratio formula for the radius of convergence

Statement

Let ∑an(x−c)n\sum a_n (x - c)^n be a real power series with an≠0a_n \ne 0 for all large nn. Assume that ∣an+1∣/∣an∣→L|a_{n+1}| / |a_n| \to L as n→∞n \to \infty, for some LL in [0,∞][0, \infty]. Then the radius of convergence of the series is R=1/LR = 1/L, with the conventions 1/0=∞1/0 = \infty and 1/∞=01/\infty = 0.

Remarks

The proof applies the ratio test to ∑an(x−c)n\sum a_n (x - c)^n. The ratio of consecutive terms tends to L∣x−c∣L |x - c|. The formula needs the limit to exist. The Cauchy–Hadamard theorem works without this assumption. For example, ∑xn/n!\sum x^n / n! has L=0L = 0 and R=∞R = \infty. Mathlib states a form for nonzero LL as FormalMultilinearSeries.ofScalars_radius_eq_inv_of_tendsto.

Explanations

L1 human

When the ratio of consecutive coefficients settles to LL, the coefficients behave like LnL^n. So the series behaves like a geometric series with ratio L∣x−c∣L |x - c|.

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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 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 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 seriesexampleRatio test, statement, no evidence levelRatio teststatementRatio formula for the radius of convergence, statement, no evidence levelRatio formula for the…statement
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  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. A series of nonnegative terms converges iff its partial sums are bounded statement
  18. Absolute convergence definition
  19. Sum of a series definition
  20. Comparison test statement
  21. Geometric series example
  22. Ratio test statement
  23. Ratio formula for the radius of convergence statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Radius_of_convergence, Theoretical radius

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