Math 2.0

example

Improper integral of x^p on an unbounded interval

Example

Let pp be a real number. The improper integral ∫1∞xp dx\int_1^\infty x^p \, dx converges if and only if p<−1p < -1. In this case, ∫1∞xp dx=−1p+1.\int_1^\infty x^p \, dx = -\frac{1}{p + 1}. For p≠−1p \ne -1, the power formula gives ∫1bxp dx=bp+1−1p+1\int_1^b x^p \, dx = \frac{b^{p+1} - 1}{p + 1}. This has a finite limit as b→∞b \to \infty exactly when p+1<0p + 1 < 0. For p=−1p = -1, ∫1bdxx=ln⁡b\int_1^b \frac{dx}{x} = \ln b, which tends to infinity.

Remarks

This example parallels the pp-series test through the integral test. Mathlib states a form as integrableOn_Ioi_rpow_iff.

Explanations

L1 human

The function xpx^p must decay faster than 1/x1/x for the area over [1,∞)[1, \infty) to be finite. The function 1/x1/x itself is just too slow.

What this node 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 node is. They do not measure its truth.

What this needs

To read this node, read these 73 nodes in this order. The foundations come first, and the list ends with this node.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Partition of an interval definition
  4. Sequence definition
  5. Strictly monotone function definition
  6. Limit of a function definition
  7. Local minimum definition
  8. Mesh of a partition definition
  9. Riemann sum definition
  10. Real sequence definition
  11. Subsequence definition
  12. Continuity at a point definition
  13. Derivative definition
  14. Limit at infinity definition
  15. One-sided limit definition
  16. Riemann integrable function definition
  17. Bounded sequence definition
  18. Geometric sequence definition
  19. Limit of a sequence definition
  20. Monotone sequence definition
  21. Series definition
  22. Continuous function definition
  23. Epsilon-delta definition of continuity statement
  24. Differentiable function definition
  25. Fermat's theorem on stationary points statement
  26. Riemann integral definition
  27. Sum of a finite geometric series statement
  28. Every sequence has a monotone subsequence statement
  29. Monotone convergence theorem for sequences statement
  30. Convergent sequence definition
  31. Squeeze theorem statement
  32. Cauchy product definition
  33. Partial sum definition
  34. Power series definition
  35. Intermediate value theorem statement
  36. Composition of continuous functions is continuous statement
  37. Differentiable functions are continuous statement
  38. Improper integral definition
  39. One-sided limits of a monotone function exist statement
  40. Bolzano–Weierstrass theorem statement
  41. The sequence r^n tends to zero for |r| < 1 example
  42. Radius of convergence definition
  43. Convergent series definition
  44. Chain rule statement
  45. Product rule statement
  46. Boundedness theorem statement
  47. Continuity of the inverse of a strictly monotone continuous function statement
  48. Absolute convergence definition
  49. A series of nonnegative terms converges iff its partial sums are bounded statement
  50. Sum of a series definition
  51. Extreme value theorem statement
  52. Power rule statement
  53. Derivative of an inverse function statement
  54. Comparison test statement
  55. Geometric series example
  56. Cauchy product of absolutely convergent series statement
  57. Rolle's theorem statement
  58. Ratio test statement
  59. Mean value theorem statement
  60. Ratio formula for the radius of convergence statement
  61. A function with positive derivative is strictly increasing statement
  62. Second fundamental theorem of calculus statement
  63. The exponential series has infinite radius of convergence statement
  64. Integral of a power statement
  65. Exponential function definition
  66. Exponential of a sum statement
  67. The exponential function is positive statement
  68. The exponential function is its own derivative statement
  69. The exponential function is strictly increasing statement
  70. Natural logarithm definition
  71. Derivative of the logarithm statement
  72. Integral of 1/x statement
  73. Improper integral of x^p on an unbounded interval example

Sources

  1. Web https://en.wikipedia.org/wiki/Improper_integral, Examples

What links here

No node links to this node yet.