Improper integral of x^p on an unbounded interval

Builds on Improper integral Integral of a power Integral of 1/x

Example

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

Remarks

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

Reading path

To understand this page from the ground up, read these 73 pages in order. The foundations come first.

Limit point of a set definition
Local maximum definition
Partition of an interval definition
Sequence definition
Strictly monotone function definition
Limit of a function definition
Local minimum definition
Mesh of a partition definition
Riemann sum definition
Real sequence definition
Subsequence definition
Continuity at a point definition
Derivative definition
Limit at infinity definition
One-sided limit definition
Riemann integrable function definition
Bounded sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Continuous function definition
Epsilon-delta definition of continuity statement
Differentiable function definition
Fermat's theorem on stationary points statement
Riemann integral definition
Sum of a finite geometric series statement
Every sequence has a monotone subsequence statement
Monotone convergence theorem for sequences statement
Convergent sequence definition
Squeeze theorem statement
Cauchy product definition
Partial sum definition
Power series definition
Intermediate value theorem statement
Composition of continuous functions is continuous statement
Differentiable functions are continuous statement
Improper integral definition
One-sided limits of a monotone function exist statement
Bolzano–Weierstrass theorem statement
The sequence r^n tends to zero for |r| < 1 example
Radius of convergence definition
Convergent series definition
Chain rule statement
Product rule statement
Boundedness theorem statement
Continuity of the inverse of a strictly monotone continuous function statement
Absolute convergence definition
A series of nonnegative terms converges iff its partial sums are bounded statement
Sum of a series definition
Extreme value theorem statement
Power rule statement
Derivative of an inverse function statement
Comparison test statement
Geometric series example
Cauchy product of absolutely convergent series statement
Rolle's theorem statement
Ratio test statement
Mean value theorem statement
Ratio formula for the radius of convergence statement
A function with positive derivative is strictly increasing statement
Second fundamental theorem of calculus statement
The exponential series has infinite radius of convergence statement
Integral of a power statement
Exponential function definition
Exponential of a sum statement
The exponential function is positive statement
The exponential function is its own derivative statement
The exponential function is strictly increasing statement
Natural logarithm definition
Derivative of the logarithm statement
Integral of 1/x statement
Improper integral of x^p on an unbounded interval example

Tag	`0VGKB2NC4T`
URI	`m2://0VGKB2NC4T`
Kind	example
Writing	W0 stub
MSC	40A10
Instance of	Improper integral
Uses	Improper integral, Integral of a power, Integral of 1/x

Improper integral of x^p on an unbounded interval

Example

Remarks

Reading path

Further reading

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