Comparison Theorem

What

Suppose $f$ and $g$ are continuous with $f(x)\ge g(x)\ge 0$ for all $x\ge a$. Then,

• If ${\int }_a^{\infty }f(x)dx$ converges, then ${\int }_a^{\infty }g(x)dx$ converges (is convergent).
• If ${\int }_a^{\infty }g(x)dx$ diverges, then ${\int }_a^{\infty }f(x)dx$ diverges (is divergent).

Similar statements hold for ${\int }_{-\infty }^{a}f(x)dx$, and so on, as well as for improper integrals of type 2.

The Comparison Theorem is useful for examples where we only want to know if ${\int }_a^{\infty }f(x)dx$ converges, since perhaps we cannot find the antiderivative of $f$.

Tip:

Think of similar integrals we know (1/x etc.) and try to compare it relative to that one.

Similar for the Limit Comparison Test of Series

Let $a_n,b_n\ge 0$ for all $n\ge 1$.

1. If ${\lim }_{n\to \infty }\frac{a_n}{b_n}=L\ne 0$, then ${\sum }_{n=1}^{\infty }{a}_n$ and ${\sum }_{n=1}^{\infty }{b}_n$ both converge or both diverge.
2. If ${\lim }_{n\to \infty }\frac{a_n}{b_n}=0$, and ${\sum }_{n=1}^{\infty }{b}_n$ converges, then ${\sum }_{n=1}^{\infty }{a}_n$ converges.
3. If ${\lim }_{n\to \infty }\frac{a_n}{b_n}=\infty$, and ${\sum }_{n=1}^{\infty }{b}_n$ diverges, then ${\sum }_{n=1}^{\infty }{a}_n$ diverges.