Math 2.0

statement theorem

Power series of the exponential function

Statement

For every real xx, ex=∑n=0∞xnn!.e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}. This series is the Maclaurin series of the exponential function. Its partial sums converge to exe^x uniformly on each bounded interval.

Remarks

By the derivative rule, every derivative of exp⁡\exp at 00 equals exp⁡(0)=1\exp(0) = 1. So the Maclaurin coefficients are 1/n!1/n!. With the series as the definition of exp⁡\exp, the first claim holds by definition. Uniform convergence follows from the result on power series. The partial sums give good approximations near 00. For example, 1+x+x2/21 + x + x^2/2 approximates exe^x with error O(x3)O(x^3).

Explanations

L1 human

The exponential function equals the sum of its Taylor polynomials at 00. Adding more terms gives better approximations of exe^x, for every xx.

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What this needs

To read this node, read these 56 nodes in this order. The foundations come first, and the list ends with this node.

  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Subsequence definition
  6. Continuity at a point definition
  7. Derivative definition
  8. Bounded sequence definition
  9. Cauchy sequence definition
  10. Geometric sequence definition
  11. Limit of a sequence definition
  12. Monotone sequence definition
  13. Series definition
  14. Continuous function definition
  15. Differentiable function definition
  16. Cauchy sequences are bounded statement
  17. Every sequence has a monotone subsequence statement
  18. Sum of a finite geometric series statement
  19. Monotone convergence theorem for sequences statement
  20. Convergent sequence definition
  21. Pointwise convergence definition
  22. Squeeze theorem statement
  23. Cauchy product definition
  24. Partial sum definition
  25. Power series definition
  26. Derivative function definition
  27. Bolzano–Weierstrass theorem statement
  28. The sequence r^n tends to zero for |r| < 1 example
  29. Radius of convergence definition
  30. Uniform convergence definition
  31. Convergent series definition
  32. Higher derivative definition
  33. Cauchy criterion for sequences statement
  34. Uniform convergence on a set definition
  35. Absolute convergence definition
  36. Series of functions definition
  37. A series of nonnegative terms converges iff its partial sums are bounded statement
  38. Sum of a series definition
  39. Taylor polynomial definition
  40. Continuously differentiable function definition
  41. Uniformly Cauchy sequence of functions definition
  42. Geometric series example
  43. Comparison test statement
  44. Cauchy product of absolutely convergent series statement
  45. Smooth function definition
  46. Ratio test statement
  47. Weierstrass M-test statement
  48. Ratio formula for the radius of convergence statement
  49. Taylor series definition
  50. Power series converge uniformly on smaller discs inside the radius of convergence statement
  51. Maclaurin series definition
  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 its own derivative statement
  56. Power series of the exponential function statement

Sources

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

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