Comparison test

Here is the Comparison Test from last time.

Suppose $f(x)\ge 0$ for all $x$. To see if $${\int }_a^{\mathrm{\infty }}f(x)𝑑x$$ converges: • First guess, by how $f(x)$ “behaves” for large $x$, whether it converges. • Then confirm: – To show the integral converges, find a $g(x)$ such that $$0\le f(x)\le g(x)$$ and $${\int }_a^{\mathrm{\infty }}g(x)𝑑x$$ converges. – To show the integral diverges, find a $g(x)$ such that $$f(x)\ge g(x)\ge 0$$ and $${\int }_a^{\mathrm{\infty }}g(x)𝑑x$$ diverges.

For integrals of the form ${\mathrm{\int }}_a^{b}f\mathbf{}\mathrm{(}x\mathrm{)}\mathbf{}𝑑x$ where $f$ has a vertical asymptote in the interval $\mathrm{[}a\mathrm{,}b\mathrm{]}$, the Comparison Test works exactly the same way.

Useful integrals for comparison: • ${\int }_1^{\mathrm{\infty }}\frac{1}{x^p}𝑑x$ converges if $p>1$ and diverges if $p\le 1$. • ${\int }_0^1\frac{1}{x^p}𝑑x$ converges if and diverges if $p\ge 1$. • ${\int }_0^{\mathrm{\infty }}{e}^{-ax}𝑑x$ for all $a>0$.