# 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 AbsoluteConvergenceOfIntegralAndBoundednessOfDerivative 2013-03-22 19:01:28 2013-03-22 19:01:28 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 40A10 NecessaryConditionOfConvergence