Math 2.0

Functions of a complex variable Theorem

Power series of arctangent

Builds on ArctangentDerivative of arctangentGeometric seriesAbel's theoremLeibniz formula for pi

Statement

For every real xx with ∣x∣≤1|x| \le 1, arctan⁡x=∑n=0∞(−1)nx2n+12n+1=x−x33+x55−⋯ .\arctan x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n + 1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots.

Remarks

For ∣x∣<1|x| < 1, integrate the geometric series 11+t2=∑(−1)nt2n\frac{1}{1 + t^2} = \sum (-1)^n t^{2n} term by term from 00 to xx. Its sum is the derivative of arctangent. At x=±1x = \pm 1, the series converges by the alternating series test, and Abel's theorem gives the equality. At x=1x = 1 this is the Leibniz formula π/4=1−1/3+1/5−⋯\pi/4 = 1 - 1/3 + 1/5 - \cdots. The radius of convergence is 11. Mathlib states the series for ∣x∣<1|x| < 1 as Real.hasSum_arctan.

Reading path

To understand this page from the ground up, read these 81 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Strictly monotone function definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Real sequence definition
  8. Subsequence definition
  9. Continuity at a point definition
  10. Derivative definition
  11. One-sided limit definition
  12. Bounded sequence definition
  13. Cauchy sequence definition
  14. Geometric sequence definition
  15. Limit of a sequence definition
  16. Monotone sequence definition
  17. Series definition
  18. Continuous function definition
  19. Epsilon-delta definition of continuity statement
  20. Differentiable function definition
  21. Fermat's theorem on stationary points statement
  22. Limit superior definition
  23. Cauchy sequences are bounded statement
  24. Every sequence has a monotone subsequence statement
  25. Sum of a finite geometric series statement
  26. Monotone convergence theorem for sequences statement
  27. Convergent sequence definition
  28. Pointwise convergence definition
  29. Squeeze theorem statement
  30. Alternating series definition
  31. Partial sum definition
  32. Power series definition
  33. Intermediate value theorem statement
  34. Composition of continuous functions is continuous statement
  35. Differentiable functions are continuous statement
  36. One-sided limits of a monotone function exist statement
  37. Bolzano–Weierstrass theorem statement
  38. The sequence r^n tends to zero for |r| < 1 example
  39. Radius of convergence definition
  40. Alternating series test statement
  41. Convergent series definition
  42. Uniform convergence definition
  43. Sine and cosine definition
  44. Chain rule statement
  45. Product rule statement
  46. Reciprocal rule statement
  47. Boundedness theorem statement
  48. Continuity of the inverse of a strictly monotone continuous function statement
  49. Cauchy criterion for sequences statement
  50. Abel's theorem statement
  51. Absolute convergence definition
  52. Series of functions definition
  53. Uniform convergence on a set definition
  54. A series of nonnegative terms converges iff its partial sums are bounded statement
  55. Tangent function definition
  56. Sum of a series definition
  57. Extreme value theorem statement
  58. Quotient rule statement
  59. Derivative of an inverse function statement
  60. Uniformly Cauchy sequence of functions definition
  61. Geometric series example
  62. Arctangent definition
  63. Comparison test statement
  64. Rolle's theorem statement
  65. Root test statement
  66. Weierstrass M-test statement
  67. Mean value theorem statement
  68. Cauchy–Hadamard theorem statement
  69. Power series converge uniformly on smaller discs inside the radius of convergence statement
  70. A function with zero derivative is constant statement
  71. Uniform convergence of derivatives allows term-by-term differentiation statement
  72. Term-by-term differentiation statement
  73. Power series are differentiable inside the radius of convergence statement
  74. Derivative of cosine statement
  75. Derivative of sine statement
  76. Pythagorean trigonometric identity statement
  77. pi object
  78. Derivative of tangent statement
  79. Leibniz formula for pi statement
  80. Derivative of arctangent statement
  81. Power series of arctangent statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_trigonometric_functions, Infinite series
About this page
Tag0T32ME0A6H
URIm2://0T32ME0A6H
Aliasesgregory-series
Kindstatement, theorem
WritingW0 stub
MSC30B10, 33B10
StatusThis copy of the site has no build data. Run m2 status 0T32ME0A6H.
UsesArctangent, Derivative of arctangent, Geometric series, Abel's theorem, Leibniz formula for pi

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.