absolute convergence of integral and boundedness of derivative

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

$\int_{a}^{\infty}\!f(x)\,dx$

where the real function $f$ and its derivative $f^{\prime}$ are continuous and $f^{\prime}$ additionally bounded on the interval $[a,\,\infty)$ . Then

$\displaystyle\lim_{x\to\infty}f(x)\;=\;0.$

(1)

Proof. If $c>a$ , we obtain

$\int_{a}^{c}\!f(x)f^{\prime}(x)\,dx\;=\;\frac{1}{2}\operatornamewithlimits{% \Big{/}}_{\!\!\!a}^{\,\quad c}\!(f(x))^{2}\;=\;\frac{(f(c))^{2}-(f(a))^{2}}{2},$

from which

$\displaystyle(f(c))^{2}\;=\;(f(a))^{2}+2\!\int_{a}^{c}\!f(x)f^{\prime}(x)\,dx.$

(2)

Using the boundedness of $f^{\prime}$ and the absolute convergence, we can estimate upwards the integral

$\int_{a}^{c}\!|f(x)f^{\prime}(x)|\,dx\;=\;\int_{a}^{c}\!|f(x)||f^{\prime}(x)|% \,dx\;\leqq\;M\!\int_{a}^{c}\!|f(x)|\,dx\;\leqq\;M\!\int_{a}^{\infty}\!|f(x)|% \,dx\quad\forall c\in[a,\,\infty)$

whence $\int_{a}^{\infty}\!|f(x)f^{\prime}(x)|\,dx$ is finite and thus $\int_{a}^{\infty}\!f(x)f^{\prime}(x)\,dx$ converges absolutely. Hence (2) implies

$\lim_{c\to\infty}(f(c))^{2}\;=\;(f(a))^{2}+2\int_{a}^{\infty}\!f(x)f^{\prime}(% x)\,dx,$

i.e. $\displaystyle\lim_{x\to\infty}(f(x))^{2}$ exists as finite, therefore also

$\lim_{x\to\infty}|f(x)|\;:=\;A.$

Antithesis: $A>0$ . It implies that there is an $x_{0}\;(\geqq a)$ such that

$|f(x)|\;\geqq\;\frac{A}{2}\quad\forall x\geqq x_{0}.$

If now $b>x_{0}$ , then we had

$\int_{x_{0}}^{b}\!|f(x)|\,dx\;\geqq\;\frac{A}{2}(b\!-\!x_{0})\;\;% \longrightarrow\infty\quad\mbox{as}\;\;b\to\infty.$

This means that $\int_{x_{0}}^{\infty}|f(x)|\,dx$ and consequently also $\int_{a}^{\infty}|f(x)|\,dx$ 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