convergence of integrals

Similarly as one speaks of convergence of series, one can speak of convergence of integrals, especially of Riemann integrals


This integralDlmfPlanetmath is convergent, if it exists, and otherwise divergent.  One can also speak of absolute convergence of integrals.

Example.  Study the convergence of the integral

∫12d⁒x(ln⁑x)c (1)

where c is a real constant.

According to the logarithm series, we may write for  1<x<b,  where b is sufficiently close to 1, the estimations


Let  1<a<b. 

1∘.  For  c>1:

∫abd⁒x(ln⁑x)c β‰§βˆ«abd⁒x2c⁒(x-1)c=-12c⁒/x=ab⁑1(c-1)⁒(x-1)c-1
 =12c⁒(c-1)⁒[1(a-1)c-1-1(b-1)c-1]βŸΆβˆžβ€ƒas aβ†’1+

2∘.  For  c=1:

∫abd⁒xln⁑x β‰§βˆ«abd⁒x2⁒(x-1)=12⁒/ab⁑ln⁑(x-1)
 =12⁒[ln⁑(b-1)-ln⁑(a-1)]βŸΆβˆžβ€ƒas aβ†’1+

3∘.  For  c<1:

0<∫abd⁒x(ln⁑x)c β‰¦βˆ«ab2c⁒d⁒x(x-1)c= 2c⁒/x=ab⁑x1-c1-c
 =2c1-c⁒[(b-1)1-c-(a-1)1-c]⟢2c1-c⁒(b-1)1-c as aβ†’1+

Consequently, the integral ∫abd⁒x(ln⁑x)c, and thus also (1), converges if and only if  c<1.

Title convergence of integrals
Canonical name ConvergenceOfIntegrals
Date of creation 2013-03-22 18:59:51
Last modified on 2013-03-22 18:59:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Example
Classification msc 40A10
Related topic UniformConvergenceOfIntegral
Related topic LogarithmicIntegral2
Related topic ListOfImproperIntegrals
Related topic SubstitutionNotation
Related topic ONotation
Defines convergent integral
Defines divergent integral