The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Comparison Test


Let c be a positive constant.

Suppose 09_infinite_series-167.gif and 09_infinite_series-168.gif are positive term

series and an ≤ cbn for all n.

(i) If 09_infinite_series-169.gif converges then 09_infinite_series-170.gif converges.

(ii) If 09_infinite_series-171.gif diverges then 09_infinite_series-172.gif diverges.


(i) Suppose 09_infinite_series-173.gif converges to S. The Constant Rule gives


Each finite partial sum of 09_infinite_series-175.gif is less than cS,


Therefore, an infinite partial sum 09_infinite_series-177.gif is less than cS and hence finite.

It follows that 09_infinite_series-178.gif converges.

(ii) If 09_infinite_series-179.gif diverges then 09_infinite_series-180.gif cannot converge by part (i).

To use the Comparison Test we compare a series whose convergence or divergence is unknown with one which is known.

Example 3: Test for Convergence
Example 4: Test for Convergence

The harmonic series 09_infinite_series-187.gif diverges, whence the given series diverges.

Sometimes the following comparison test is easier to use.

Last Update: 2006-11-07