Elementary Calculus: Comparison Tests for Absolute Convergence

Comparison Tests for Absolute Convergence

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

COMPARISON TEST

If |a_n| ≤ c|b_n| and is absolutely convergent then is absolutely convergent.

LIMIT COMPARISON TEST

Let c be a positive real number. If

|a_K| ≤ c|b_K| for all infinite K and is absolutely convergent then 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.

RATIO TEST

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

(i) If L < 1, the series 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.

PROOF

(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 = |a_N|.

|a_N+l| < cb, |a_N ₊ ₂| < cb²,...,|a_N ₊ _n| < cb",....

The geometric series converges, so by the Comparison Test, the tail converges. Therefore the absolute value series converges.

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

are all greater than one. Therefore

|a_N| < |a_N+1| < ... < |a_N+n| < ....

It follows that the terms a_n do not converge to zero, so the series 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)