Math 2.0

Approximations and expansions Asymptotic approximations and expansions Theorem

Stirling's formula

Builds on Asymptotic equivalenceepiWallis product

Statement

As n→∞n \to \infty, n!∼2πn(ne)n.n! \sim \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n. That is, the ratio n!2πn (n/e)n\frac{n!}{\sqrt{2 \pi n} \, (n/e)^n} tends to 11.

Remarks

The relation ∼\sim is asymptotic equivalence. The absolute error grows, but the relative error tends to 00. Comparing ln⁡n!=∑k≤nln⁡k\ln n! = \sum_{k \le n} \ln k with ∫1nln⁡x dx\int_1^n \ln x \, dx gives the form n!∼Cn(n/e)nn! \sim C \sqrt{n} (n/e)^n. The Wallis product gives C=2πC = \sqrt{2\pi}. A sharper form is n!=2πn(n/e)n(1+112n+O(n−2))n! = \sqrt{2 \pi n} (n/e)^n \big(1 + \frac{1}{12 n} + O(n^{-2})\big). Mathlib states the limit as Stirling.tendsto_stirlingSeq_sqrt_pi.

History

Abraham de Moivre first stated a less precise form of the result. In 1730, James Stirling showed that the constant is 2π\sqrt{2\pi}.

Reading path

To understand this page from the ground up, read these 79 pages in order. The foundations come first.

  1. Big O notation definition
  2. Limit point of a set definition
  3. Local maximum definition
  4. Partition of an interval definition
  5. Sequence definition
  6. Little o notation definition
  7. Limit of a function definition
  8. Local minimum definition
  9. Mesh of a partition definition
  10. Riemann sum definition
  11. Real sequence definition
  12. Subsequence definition
  13. Asymptotic equivalence definition
  14. Continuity at a point definition
  15. Derivative definition
  16. Riemann integrable function definition
  17. Bounded sequence definition
  18. Cauchy sequence definition
  19. Geometric sequence definition
  20. Limit of a sequence definition
  21. Monotone sequence definition
  22. Series definition
  23. Continuous function definition
  24. Differentiable function definition
  25. Fermat's theorem on stationary points statement
  26. Limit superior definition
  27. Cauchy sequences are bounded statement
  28. Sum of a finite geometric series statement
  29. Monotone convergence theorem for sequences statement
  30. Convergent sequence definition
  31. Every sequence has a monotone subsequence statement
  32. Pointwise convergence definition
  33. Squeeze theorem statement
  34. Partial sum definition
  35. Power series definition
  36. Intermediate value theorem statement
  37. Differentiable functions are continuous statement
  38. Derivative function definition
  39. Bolzano–Weierstrass theorem statement
  40. The sequence r^n tends to zero for |r| < 1 example
  41. Radius of convergence definition
  42. Convergent series definition
  43. Uniform convergence definition
  44. Sine and cosine definition
  45. Boundedness theorem statement
  46. Product rule statement
  47. Higher derivative definition
  48. Cauchy criterion for sequences statement
  49. Uniform convergence on a set definition
  50. Absolute convergence definition
  51. A series of nonnegative terms converges iff its partial sums are bounded statement
  52. Series of functions definition
  53. Sum of a series definition
  54. Extreme value theorem statement
  55. Continuously differentiable function definition
  56. Uniformly Cauchy sequence of functions definition
  57. Geometric series example
  58. Comparison test statement
  59. Rolle's theorem statement
  60. Root test statement
  61. Ratio test statement
  62. Weierstrass M-test statement
  63. Mean value theorem statement
  64. Cauchy–Hadamard theorem statement
  65. Ratio formula for the radius of convergence statement
  66. Power series converge uniformly on smaller discs inside the radius of convergence statement
  67. Second fundamental theorem of calculus statement
  68. Uniform convergence of derivatives allows term-by-term differentiation statement
  69. The exponential series has infinite radius of convergence statement
  70. Integration by parts statement
  71. Term-by-term differentiation statement
  72. Exponential function definition
  73. Reduction formula concept
  74. e object
  75. Power series are differentiable inside the radius of convergence statement
  76. Derivative of cosine statement
  77. Wallis product statement
  78. pi object
  79. Stirling's formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Stirling%27s_approximation, Derivation
About this page
Tag0KC0F51T6A
URIm2://0KC0F51T6A
Aliasesstirling-approximation
Kindstatement, theorem
WritingW1 stated
MSC41A60, 05A10
StatusThis copy of the site has no build data. Run m2 status 0KC0F51T6A.
UsesAsymptotic equivalence, e, pi, Wallis product
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