Barrons AP Calculus

Figure  N7–26

THE COMPARISON TEST

We can often determine whether an improper integral converges or diverges by
comparing it to a known integral on the same interval. This method is especially
helpful when it is not easy to actually evaluate the appropriate limit by finding
an antiderivative for the integrand. There are two cases.

(1) Convergence. If on the interval of integration f (x) ≤ g(x) and is

known    to  converge,   then    also    converges.  For     example,    consider   

. We know that converges. Since , the improper
integral must also converge.

(2) Divergence. If on the interval of integration f (x) ≥ g(x) and is known

to  diverge,    then        also    diverges.   For example,    consider     .  We

know    that        diverges.   Since   sec x   ≥   1,  it  follows that     ;  hence

the improper    integral        must    also    diverge.

BC  ONLY