Comparison Tests for Convergence

Suppose that $f(x)$ and $g(x)$ are functions on $[a,\infty)$, and that $$0 \le f(x) \le g(x) \quad\text{ for all }x \in [a,\infty).$$Then we say $f$ is bounded between $0$ and $g$.  Observe the following, based on the inequality above:
  • If $\int_a^\infty g(x)\, dx$ converges, then $\int_a^\infty f(x)\, dx$ must also converge.

  • If $\int_a^\infty f(x)\, dx$ diverges, then $\int_a^\infty g(x)\, dx$ must also diverge.

  • If $\int_a^\infty f(x) \,dx$ converges, we know nothing about $\int_a^\infty g(x) \,dx$ - it might converge or diverge.

  • If $\int_a^\infty g(x) \,dx$ diverges, we know nothing about $\int_a^\infty f(x) \,dx$ - it might converge or diverge.

Example: $\displaystyle\int_1^\infty \frac{dx}{x^4}$ converges, and, since $\displaystyle\frac{1}{x^4+1} < \frac{1}{x^4}$, we can conclude that also $\displaystyle\int_1^\infty \frac{dx}{1+x^4}$ converges. Notice that the test doesn't tell us the value of the integral, just that it converges.

The idea of comparison tests will be used often when we get to sequences and series. It's a good idea to pay close attention to the following video, and get a feel for comparison tests now.