a sufficient condition for convergence of integral

Suppose that the real function f is positive and continuousMathworldPlanetmath on the interval[a,).  A sufficient condition for the convergence (http://planetmath.org/ConvergentIntegral) of the improper integral

af(x)𝑑x (1)

is that

limxf(x+1)f(x)=q< 1. (2)

Proof.  Assume that the condition (2) is in .  For an indirect proof (http://planetmath.org/ReductioAdAbsurdum), make the antithesis that the integral (http://planetmath.org/RiemannIntegral) (1) diverges (http://planetmath.org/DivergentIntegral).

Because of the positiveness, we have  af(x)𝑑x=.  We can use l’Hôpital’s rule (http://planetmath.org/LHpitalsRule):


Using the http://planetmath.org/node/11373substitution  x+1=t  we get


and dividing this equation by acf(t)𝑑t and taking limits (http://planetmath.org/ImproperLimits) yield (f is bounded!)

1>q=limcacf(x+1)𝑑xacf(x)𝑑x= 0+1-0= 1.

This contradictory result shows that the antithesis is wrong; thus (1) must be convergentMathworldPlanetmathPlanetmath (http://planetmath.org/ConvergentIntegral).

Note.  The condition (2) is not necessary for the convergence of (1).  This is seen e.g. in the case of the converging of (2) equals 1.

Title a sufficient condition for convergence of integral
Canonical name ASufficientConditionForConvergenceOfIntegral
Date of creation 2013-03-22 19:01:13
Last modified on 2013-03-22 19:01:13
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 40A10
Related topic RatioTest