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absolute convergence of integral and boundedness of derivative


Theorem.  Assume that we have an http://planetmath.org/node/11865absolutely converging integralDlmfPlanetmath

∫∞af(x)𝑑x

where the real function f and its derivative fβ€² are continuousMathworldPlanetmath and fβ€² additionally bounded on the interval  [a,∞).  Then

lim (1)

Proof.  If  c>a,  we obtain

∫acf(x)fβ€²(x)dx=12/ac(f(x))2=(f⁒(c))2-(f⁒(a))22,

from which

(f⁒(c))2=(f⁒(a))2+2⁒∫acf⁒(x)⁒f′⁒(x)⁒𝑑x. (2)

Using the boundedness of fβ€² and the absolute convergenceMathworldPlanetmath, we can estimate upwards the integral

∫ac|f⁒(x)⁒f′⁒(x)|⁒𝑑x=∫ac|f⁒(x)|⁒|f′⁒(x)|⁒𝑑x≦M⁒∫ac|f⁒(x)|⁒𝑑x≦M⁒∫a∞|f⁒(x)|⁒𝑑xβ€ƒβˆ€c∈[a,∞)

whence ∫a∞|f⁒(x)⁒f′⁒(x)|⁒𝑑x is finite and thus ∫a∞f⁒(x)⁒f′⁒(x)⁒𝑑x converges absolutely.  Hence (2) implies

limcβ†’βˆžβ‘(f⁒(c))2=(f⁒(a))2+2⁒∫a∞f⁒(x)⁒f′⁒(x)⁒𝑑x,

i.e. limxβ†’βˆžβ‘(f⁒(x))2 exists as finite, therefore also

limxβ†’βˆžβ‘|f⁒(x)|:=A.

Antithesis:  A>0.  It implies that there is an x0(≧a) such that

|f⁒(x)|≧A2β€ƒβˆ€x≧x0.

If now  b>x0, then we had

∫x0b|f⁒(x)|⁒𝑑x≧A2⁒(b-x0)βŸΆβˆžβ€ƒas⁒bβ†’βˆž.

This means that ∫x0∞|f⁒(x)|⁒𝑑x and consequently also ∫a∞|f⁒(x)|⁒𝑑x would be divergent.  Since it is not true, we infer that  A=0,  i.e. that the assertion (1) is true.

Title absolute convergence of integral and boundedness of derivative
Canonical name AbsoluteConvergenceOfIntegralAndBoundednessOfDerivative
Date of creation 2013-03-22 19:01:28
Last modified on 2013-03-22 19:01:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 40A10
Related topic NecessaryConditionOfConvergence