Extension: Integrand Divergence Test for Improper Integrals

Improper Integrals with unbounded intervals of integration

Now that you know what an improper integral is, you should learn how to test of these integrals actually converge. If an integral converges, its limits exists and approaches a finite number. One of the easiest ways to check for convergence is to see what happens to the integrand as $x\to\infty$. Let $f(x)$ be Riemann integrable on $[a,\infty]$. Then $$\lim_{b\to\infty}\int_a^b f(x)\,dx<\infty \implies \lim_{x\to\infty} f(x)=0$$ Notice that the converse is not necessarily true. Take $f(x)=\frac{1}{x}$. Then $\lim_{x\to\infty} f(x)=0$ but the integral does not converge for any $a>0$: $$\lim_{b\to\infty}\int_a^b \frac{1}{x}\,dx=\lim_{b\to\infty} \ln\left(\frac{b}{a}\right)=\infty$$ Although the integrand approached zero, the improper integral did not exist.

Remark: There is one caveat here: this test only holds for uniformly continuous functions. For example, see https://en.wikipedia.org/wiki/Fresnel_integral