Comparison test

Definition

The Comparison Test is used to determine the convergence or divergence of an infinite series by comparing it to another series with known behavior. It involves either the Direct Comparison Test or the Limit Comparison Test.

5 Must Know Facts For Your Next Test

1. The Direct Comparison Test requires comparing a given series to a second series that is known to converge or diverge.
2. For the Direct Comparison Test, if $0 \leq a_n \leq b_n$ for all $n$ and $\sum b_n$ converges, then $\sum a_n$ also converges.
3. Conversely, if $0 \leq b_n \leq a_n$ for all $n$ and $\sum b_n$ diverges, then $\sum a_n$ also diverges.
4. The Limit Comparison Test involves taking the limit of the ratio of the terms of two series: if $\lim_{{n \to \infty}} \frac{a_n}{b_n} = c > 0$, where $c$ is finite and positive, then both series either converge or diverge together.
5. Comparison tests are particularly useful when dealing with positive term series.

Review Questions

• How do you apply the Direct Comparison Test to determine if a series converges?
• What does it mean if the limit in the Limit Comparison Test equals zero?
• Why might one choose to use a comparison test instead of other methods?

Related terms

Convergence:

A property of a series where its partial sums approach a finite limit as more terms are added.

Divergence:

A property of a series where its partial sums do not approach any finite limit.

$p$-Series: $\sum_{n=1}^{\infty} \frac{1}{n^p}$, which converges if and only if $p > 1$. Often used in comparison tests.