Section Summary: 12.3

Definitions

Theorems

Suppose f is a continuous, positive, decreasing function on and let . Then the series

is convergent if and only if the improper integral

is convergent. If I diverges, then s diverges (and vice versa).

Note that it is not essential for f to be positive and decreasing everywhere, but it must be ultimately positive and decreasing (that is, decreasing beyond some fixed value of x).

The p-series

is convergent if p>1, and divergent if .

If converges by the integral test, and the remainder , then

(this gives us a bound on the error we're making in the calculation of a series).

Properties, Hints, etc.

Summary

Integrals serve as useful tools for evaluating series, determining whether series exist, and estimating the error we're making in an estimate of the limit of a series.