Math 2.0

statement proposition

The exponential series has infinite radius of convergence

Statement

The radius of convergence of the power series ∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{x^n}{n!} is ∞\infty. So the series converges absolutely for every real xx.

Remarks

The ratio of consecutive coefficients is 1/(n+1)!1/n!=1n+1→0\frac{1/(n+1)!}{1/n!} = \frac{1}{n + 1} \to 0. The ratio formula then gives radius ∞\infty. The same holds for the series of sine, cosine, sinh⁡\sinh and cosh⁡\cosh. Mathlib states it as NormedSpace.expSeries_radius_eq_top.

Explanations

L1 human

The factorials in the denominators grow faster than every power xnx^n. So the terms of the series go to 00 fast, for each fixed xx.

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To read this node, read these 24 nodes in this order. The foundations come first, and the list ends with this node.

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…statementThe exponential series has infinite radius of convergence, statement, no evidence levelThe exponential serie…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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
  24. The exponential series has infinite radius of convergence statement

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43 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Power series

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