The Integral Test

The Integral Test provides a method of proving the sum of a series to be either divergent or convergent and works in the following way.

Consider a decreasing functionSinceis decreasing,

Setthen the above becomesSum this expression fromtoto giveand sinceis convergent if and only if is bounded, and the integralconverges if and only if the sumis bounded.

We may state the Integral Test: The seriesis convergent is and only if the integralis bounded where

Example: Provedoes not converge.

and assodoes not converge.

Example: Proveconverges.