Math 2.0

Special functions Theorem

Power series of cosine

Builds on Sine and cosineRatio formula for the radius of convergenceMaclaurin series

Statement

For every real xx, the series ∑n=0∞(−1)nx2n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} converges absolutely, and cos⁡x=1−x22!+x44!−⋯ .\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots. This series is the Maclaurin series of the cosine function.

Remarks

The ratio of the terms is x2(2n+1)(2n+2)→0\frac{x^2}{(2n+1)(2n+2)} \to 0. So the radius of convergence is ∞\infty. For ∣x∣≤1|x| \le 1, the terms decrease in size. So the partial sums lie alternately above and below cos⁡x\cos x. Mathlib states the series as Real.hasSum_cos.

Reading path

To understand this page from the ground up, read these 38 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 sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded 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 levelSeriesdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionSum 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 seriesdefinitionDerivative function, definition, no evidence levelDerivative functiondefinitionThe 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 seriesdefinitionSine and cosine, definition, no evidence levelSine and cosinedefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionA 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 seriesdefinitionTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionContinuously differentiable function, definition, no evidence levelContinuously differen…definitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleSmooth function, definition, no evidence levelSmooth functiondefinitionRatio test, statement, no evidence levelRatio teststatementRatio formula for the radius of convergence, statement, no evidence levelRatio formula for the…statementTaylor series, definition, no evidence levelTaylor seriesdefinitionMaclaurin series, definition, no evidence levelMaclaurin seriesdefinitionPower series of cosine, statement, no evidence levelPower series of cosinestatement
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.
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  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Continuity at a point definition
  6. Derivative definition
  7. Bounded sequence definition
  8. Geometric sequence definition
  9. Limit of a sequence definition
  10. Monotone sequence definition
  11. Series definition
  12. Continuous function definition
  13. Differentiable function definition
  14. Sum of a finite geometric series statement
  15. Monotone convergence theorem for sequences statement
  16. Convergent sequence definition
  17. Squeeze theorem statement
  18. Partial sum definition
  19. Power series definition
  20. Derivative function definition
  21. The sequence r^n tends to zero for |r| < 1 example
  22. Radius of convergence definition
  23. Convergent series definition
  24. Sine and cosine definition
  25. Higher derivative definition
  26. A series of nonnegative terms converges iff its partial sums are bounded statement
  27. Absolute convergence definition
  28. Sum of a series definition
  29. Taylor polynomial definition
  30. Continuously differentiable function definition
  31. Comparison test statement
  32. Geometric series example
  33. Smooth function definition
  34. Ratio test statement
  35. Ratio formula for the radius of convergence statement
  36. Taylor series definition
  37. Maclaurin series definition
  38. Power series of cosine statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Sine_and_cosine, Series and polynomials
About this page
Tag08W5RN375X
URIm2://08W5RN375X
Kindstatement, theorem
WritingW0 stub
MSC33B10, 30B10
StatusThis copy of the site has no build data. Run m2 status 08W5RN375X.
UsesSine and cosine, Ratio formula for the radius of convergence, Maclaurin series

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