Math 2.0

Sequences, series, summability Example

Integral of exp(-x) over the half line

Builds on Improper integralIntegral of the exponential function

Example

The improper integral of e−xe^{-x} over [0,∞)[0, \infty) converges, and ∫0∞e−x dx=1.\int_0^\infty e^{-x} \, dx = 1. For b>0b > 0, the exponential formula with c=−1c = -1 gives ∫0be−x dx=1−e−b\int_0^b e^{-x} \, dx = 1 - e^{-b}. This tends to 11 as b→∞b \to \infty.

Remarks

It is the case s=1s = 1 of the gamma function: Γ(1)=1\Gamma(1) = 1. Mathlib states it as integral_exp_neg_Ioi_zero.

Reading path

To understand this page from the ground up, read these 55 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Partition of an interval definition
  4. Sequence definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Mesh of a partition definition
  8. Riemann sum definition
  9. Real sequence definition
  10. Subsequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. Limit at infinity definition
  14. One-sided limit definition
  15. Riemann integrable function definition
  16. Bounded sequence definition
  17. Geometric sequence definition
  18. Limit of a sequence definition
  19. Monotone sequence definition
  20. Series definition
  21. Continuous function definition
  22. Fermat's theorem on stationary points statement
  23. Riemann integral definition
  24. Sum of a finite geometric series statement
  25. Every sequence has a monotone subsequence statement
  26. Monotone convergence theorem for sequences statement
  27. Convergent sequence definition
  28. Squeeze theorem statement
  29. Cauchy product definition
  30. Partial sum definition
  31. Power series definition
  32. Improper integral definition
  33. Bolzano–Weierstrass theorem statement
  34. The sequence r^n tends to zero for |r| < 1 example
  35. Radius of convergence definition
  36. Convergent series definition
  37. Boundedness theorem statement
  38. A series of nonnegative terms converges iff its partial sums are bounded statement
  39. Absolute convergence definition
  40. Sum of a series definition
  41. Extreme value theorem statement
  42. Comparison test statement
  43. Geometric series example
  44. Cauchy product of absolutely convergent series statement
  45. Rolle's theorem statement
  46. Ratio test statement
  47. Mean value theorem statement
  48. Ratio formula for the radius of convergence statement
  49. The exponential series has infinite radius of convergence statement
  50. Second fundamental theorem of calculus statement
  51. Exponential function definition
  52. Exponential of a sum statement
  53. The exponential function is its own derivative statement
  54. Integral of the exponential function statement
  55. Integral of exp(-x) over the half line example

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Improper_integral, Convergence of the integral
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Instance ofImproper integral
UsesImproper integral, Integral of the exponential function

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