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Section 1.25 Improper integrals: convergence tests
Theorem 1.25.1 . Direct comparison test.
Let \(f\) and \(g\) be nonnegative functions on an interval \(I\text{,}\) and suppose \(f(x)\leq g(x)\) for all \(x\) in \(I\text{.}\) If the integral of \(g\) over \(I\) converges, then the integral of \(f\) over \(I\) converges. Using logical notation:
\begin{equation}
\begin{array}{c} \text{ integral of \(g\) over } \\ \text{\(I\) converges} \end{array} \implies \begin{array}{c} \text{ integral of \(f\) over} \\ \text{\(I\) converges} \end{array}\text{.}\tag{1.25.1}
\end{equation}
Equivalently, using the contrapositive of
(1.25.1) , we have
\begin{equation}
\begin{array}{c} \text{ integral of \(f\) over} \\ \text{\(I\) diverges} \end{array} \implies \begin{array}{c} \text{ integral of \(g\) over} \\ \text{\(I\) diverges} \end{array}\text{.}\tag{1.25.2}
\end{equation}
Theorem 1.25.2 . Limit comparison test.
Let \(f\) and \(g\) be continuous and positive on the interval \(I\text{.}\)
If \(I=[a,\infty)\) and \(\displaystyle\lim_{x\to\infty}\frac{f(x)}{g(x)}=L\) with \(0\lt L \lt \infty\text{,}\) then
\begin{equation*}
\int_a^\infty f(x)\, dx \text{ converges } \iff \int_a^\infty g(x)\, dx \text{ converges }\text{.}
\end{equation*}
If \(I=(-\infty,a]\) and \(\displaystyle\lim_{x\to-\infty}\frac{f(x)}{g(x)}=L\) with \(0\lt L \lt \infty\text{,}\) then
\begin{equation*}
\int_{-\infty}^a f(x)\, dx \text{ converges } \iff \int_{-\infty}^a g(x)\, dx \text{ converges }\text{.}
\end{equation*}
If \(I=(a,b]\) and \(\displaystyle\lim_{x\to a^+}\frac{f(x)}{g(x)}=L\) with \(0\lt L \lt \infty\text{,}\) then
\begin{equation*}
\int_{a}^b f(x)\, dx \text{ converges } \iff \int_a^b g(x)\, dx \text{ converges }\text{.}
\end{equation*}
If \(I=[a,b)\) and \(\displaystyle\lim_{x\to b^-}\frac{f(x)}{g(x)}=L\) with \(0\lt L \lt \infty\text{,}\) then
\begin{equation*}
\int_{a}^b f(x)\, dx \text{ converges } \iff \int_a^b g(x)\, dx \text{ converges }\text{.}
\end{equation*}
Example 1.25.3 . Comparison test.
Decide whether \(\displaystyle\int_2^\infty \frac{1}{x^5+\sqrt{x+3}}\, dx\) converges.
Example 1.25.4 . Comparison test.
Decide whether \(\displaystyle\int_1^\infty \frac{2+\sin x}{x}\, dx\) converges.
Example 1.25.5 . Comparison test.
Decide whether \(\displaystyle\int_0^\infty\frac{1}{\sqrt{x}+3x^5}\, dx\) converges.
Example 1.25.6 . Comparison test.
Let \(f(x)=ax^2+bx+c\) be any fixed irreducible quadratic polynomial with \(a>0\text{.}\) Decide whether \(\displaystyle\int_{-\infty}^\infty \frac{1}{f(x)}\, dx\) exists.
