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

Comparison Tests for Absolute Convergence

The comparison tests for positive term series give us tests for absolute convergence.


If |an| ≤ c|bn| and 09_infinite_series-346.gif is absolutely convergent then 09_infinite_series-347.gif is absolutely convergent.


Let c be a positive real number. If

|aK| ≤ c|bK| for all infinite K and 09_infinite_series-348.gif is absolutely convergent then 09_infinite_series-349.gif is absolutely convergent.

The above tests do not help to distinguish between conditional convergence and divergence. Theorem 2 in Section 9.2 is often useful as a test for divergence.

There is another test which can be used either to show that a series is absolutely convergent or that a series is divergent.


Suppose the limit of the ratio |an+1|/|an| exists or is ∞,


(i) If L < 1, the series 09_infinite_series-351.gif converges absolutely.

(ii) If L > 1, or L = ∞, the series diverges.

(iii) If L = 1, the test gives no information and the series may converge absolutely, converge conditionally, or diverge.


(i) Choose b with L < b < 1. By the ε, N condition, there is an N such that all the ratios


are less than b. Therefore with c = |aN|.

|aN+l| < cb, |aN + 2| < cb2,...,|aN + n| < cb",....

The geometric series 09_infinite_series-353.gif converges, so by the Comparison Test, the tail 09_infinite_series-354.gif converges. Therefore the absolute value series 09_infinite_series-355.gif converges.

(ii) By the ε, N condition there is an N such that the ratios


are all greater than one. Therefore

|aN| < |aN+1| < ... < |aN+n| < ....

It follows that the terms an do not converge to zero, so the series 09_infinite_series-357.gif diverges.

The Ratio Test is useful even for positive term series, and is often effective for series involving n! and an.

Example 3: Ratio Test
Example 4: Ratio Test (Diverging Series)
Example 5: Ratio Rest (Inapplicable)

Last Update: 2006-11-07