Limit comparison test

Definition

The limit comparison test is a method to determine the convergence or divergence of an infinite series by comparing it to another series with known behavior. It involves taking the limit of the ratio of terms from two different series.

5 Must Know Facts For Your Next Test

1. The limit comparison test requires both series to have positive terms.
2. If $\lim_{{n \to \infty}} \frac{{a_n}}{{b_n}} = c$ where $0 < c < \infty$, then both series either converge or diverge together.
3. The test is particularly useful when the direct comparison test is inconclusive due to oscillation or complexity.
4. Choosing a good comparison series $b_n$ that closely resembles $a_n$ is crucial for applying the test effectively.
5. The limit comparison test can be applied even if individual terms are not easily simplified, as long as their ratio approaches a finite non-zero limit.

Review Questions

• What condition must be met for the limit comparison test to be applicable?
• How do you determine whether two series converge or diverge using the limit comparison test?
• Why might one choose to use the limit comparison test over the direct comparison test?

Related terms

Direct Comparison Test: A method used to determine convergence or divergence of an infinite series by directly comparing it term-by-term with another known convergent or divergent series.

$p$-Series: A series of the form $\sum \frac{1}{n^p}$ where its convergence depends on the value of $p$. Specifically, it converges if and only if $p > 1$.

$\infty$: \text{Represents infinity, often used in limits and summations involving infinite sequences and series.}