# convergence of integrals

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

 $\int_{I}f(t)\,dt.$

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

Example.  Study the convergence of the integral

 $\displaystyle\int_{1}^{2}\frac{dx}{(\ln{x})^{c}}$ (1)

where $c$ is a real constant.

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

 $\displaystyle\ln(x\!-\!1)\;=\;x-1+O((x\!-\!1)^{2})\;=\;(x\!-\!1)[1+O(x\!-\!1)]% \;\begin{cases}\leq 2(x\!-\!1),\\ \geq\frac{1}{2}(x\!-\!1).\end{cases}$

Let  $1.

$1^{\circ}$.  For  $c>1$:

 $\displaystyle\int_{a}^{b}\frac{dx}{(\ln{x})^{c}}$ $\displaystyle\geqq\int_{a}^{b}\frac{dx}{2^{c}(x\!-\!1)^{c}}\;=\;-\frac{1}{2^{c% }}\!\operatornamewithlimits{\Big{/}}_{\!\!\!x=a}^{\,\quad b}\!\frac{1}{(c\!-\!% 1)(x\!-\!1)^{c-1}}$ $\displaystyle\;=\;\frac{1}{2^{c}(c\!-\!1)}\left[\frac{1}{(a\!-\!1)^{c-1}}-% \frac{1}{(b\!-\!1)^{c-1}}\right]\;\longrightarrow\infty\quad\mbox{as}\quad a% \to 1+$

$2^{\circ}$.  For  $c=1$:

 $\displaystyle\int_{a}^{b}\frac{dx}{\ln{x}}$ $\displaystyle\geqq\int_{a}^{b}\frac{dx}{2(x\!-\!1)}\;=\;\frac{1}{2}\!% \operatornamewithlimits{\Big{/}}_{\!\!\!a}^{\,\quad b}\!\ln(x\!-\!1)$ $\displaystyle\;=\;\frac{1}{2}\left[\ln(b\!-\!1)-\ln(a\!-\!1)\right]\;% \longrightarrow\infty\quad\mbox{as}\quad a\to 1+$

$3^{\circ}$.  For  $c<1$:

 $\displaystyle 0\;<\;\int_{a}^{b}\frac{dx}{(\ln{x})^{c}}$ $\displaystyle\leqq\int_{a}^{b}\frac{2^{c}\,dx}{(x\!-\!1)^{c}}\;=\;2^{c}\!% \operatornamewithlimits{\Big{/}}_{\!\!\!x=a}^{\,\quad b}\!\frac{x^{1-c}}{1\!-% \!c}$ $\displaystyle\;=\;\frac{2^{c}}{1\!-\!c}\left[(b\!-\!1)^{1-c}-(a\!-\!1)^{1-c}% \right]\;\longrightarrow\frac{2^{c}}{1\!-\!c}(b\!-\!1)^{1-c}\quad\mbox{as}% \quad a\to 1+$

Consequently, the integral $\int_{a}^{b}\frac{dx}{(\ln{x})^{c}}$, 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