Limit Comparison Test

Definition

The Limit Comparison Test is a method used to determine the convergence or divergence of an infinite series by comparing it to a known benchmark series. If two series are compared and the limit of their ratios is a positive finite number, both series will either converge or diverge together. This test is particularly useful for handling series that may be difficult to analyze directly, especially as they approach infinity.

5 Must Know Facts For Your Next Test

1. The Limit Comparison Test can only be applied when both series involved are positive term series.
2. To use the test, you find the limit $$L = \lim_{n \to \infty} \frac{a_n}{b_n}$$, where $$a_n$$ is the term from the series in question and $$b_n$$ is the term from a known series.
3. If the limit L is a positive finite number (i.e., 0 < L < $$\infty$$), then both series either converge or both diverge.
4. The test does not specify what happens if L equals 0 or if L equals infinity; in those cases, different methods must be applied.
5. This test is particularly effective for comparing complex series with simpler ones, like comparing an unknown series to a P-Series.

Review Questions

• How does the Limit Comparison Test determine the behavior of an infinite series?
  • The Limit Comparison Test determines the behavior of an infinite series by comparing it to another known series. When calculating the limit of their ratios, if that limit results in a positive finite number, it indicates that both series will either converge or diverge together. This means that understanding one series allows us to infer the behavior of the other, simplifying the analysis of potentially complex series.
• What are the necessary conditions for applying the Limit Comparison Test between two series?
  • To apply the Limit Comparison Test, both series being compared must consist of positive terms. Additionally, you must be able to compute the limit of their ratio as n approaches infinity. This ensures that any conclusions drawn about convergence or divergence are valid. If these conditions are not met, the test cannot be utilized effectively.
• Evaluate how using the Limit Comparison Test with a P-Series can facilitate understanding of more complex infinite series.
  • Using the Limit Comparison Test with a P-Series allows for a straightforward comparison when analyzing more complex infinite series. Since P-Series have well-defined convergence criteria based on their exponent p, they serve as effective benchmarks. By establishing a positive finite limit when comparing a complex series to a P-Series, one can easily determine whether that complex series converges or diverges, thus simplifying many convergence tests.

Related terms

Convergence: The property of a series where the sum of its terms approaches a specific value as more terms are added.

Divergence: The behavior of a series where the sum of its terms does not approach a finite limit as more terms are added.

P-Series: A specific type of series in the form $$ ext{1/n^p}$$, which converges if $$p > 1$$ and diverges if $$p \leq 1$$.