Math 2.0

Functions of a complex variable Theorem

Mercator series

Builds on Natural logarithmDerivative of the logarithmGeometric seriesAbel's theorem

Statement

For every real xx with −1<x≤1-1 < x \le 1, ln⁡(1+x)=∑n=1∞(−1)n+1nxn=x−x22+x33−⋯ .\ln(1 + x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} x^n = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots.

Remarks

For ∣x∣<1|x| < 1, integrate the geometric series 11+t=∑(−t)n\frac{1}{1 + t} = \sum (-t)^n term by term from 00 to xx. At x=1x = 1 the series converges by the alternating series test. Abel's theorem gives ln⁡2=1−12+13−⋯\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \cdots. Mathlib states the form ∑n≥0xn+1n+1=−ln⁡(1−x)\sum_{n \ge 0} \frac{x^{n+1}}{n+1} = -\ln(1 - x) as Real.hasSum_pow_div_log_of_abs_lt_one.

History

Johannes Hudde found the series in 1656 and Isaac Newton in 1665, but neither of them published it. Nicholas Mercator found it independently and used it in his book Logarithmotechnia of 1668.

Reading path

To understand this page from the ground up, read these 60 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. Geometric sequence definition
  14. Limit of a sequence definition
  15. Monotone sequence definition
  16. Series definition
  17. Continuous function definition
  18. Epsilon-delta definition of continuity statement
  19. Fermat's theorem on stationary points statement
  20. Sum of a finite geometric series statement
  21. Every sequence has a monotone subsequence statement
  22. Monotone convergence theorem for sequences statement
  23. Convergent sequence definition
  24. Squeeze theorem statement
  25. Cauchy product definition
  26. Partial sum definition
  27. Power series definition
  28. Intermediate value theorem statement
  29. Composition of continuous functions is continuous statement
  30. One-sided limits of a monotone function exist statement
  31. Bolzano–Weierstrass theorem statement
  32. The sequence r^n tends to zero for |r| < 1 example
  33. Radius of convergence definition
  34. Convergent series definition
  35. Chain rule statement
  36. Boundedness theorem statement
  37. Continuity of the inverse of a strictly monotone continuous function statement
  38. Abel's theorem statement
  39. Absolute convergence definition
  40. A series of nonnegative terms converges iff its partial sums are bounded statement
  41. Sum of a series definition
  42. Extreme value theorem statement
  43. Derivative of an inverse function statement
  44. Geometric series example
  45. Comparison test statement
  46. Cauchy product of absolutely convergent series statement
  47. Rolle's theorem statement
  48. Ratio test statement
  49. Mean value theorem statement
  50. Ratio formula for the radius of convergence statement
  51. A function with positive derivative is strictly increasing statement
  52. The exponential series has infinite radius of convergence statement
  53. Exponential function definition
  54. Exponential of a sum statement
  55. The exponential function is positive statement
  56. The exponential function is its own derivative statement
  57. The exponential function is strictly increasing statement
  58. Natural logarithm definition
  59. Derivative of the logarithm statement
  60. Mercator series statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Mercator_series, Derivation
About this page
Tag0YZXNE3HFR
URIm2://0YZXNE3HFR
Kindstatement, theorem
WritingW1 stated
MSC30B10, 33B10
StatusThis copy of the site has no build data. Run m2 status 0YZXNE3HFR.
UsesNatural logarithm, Derivative of the logarithm, Geometric series, Abel's theorem
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