Limit Comparison Calculator
The Limit Comparison Calculator is design to assist in the analysis of infinite series by comparing them to known series types, like p-series or geometric series. This comparison helps determine the behavior of the series under investigation, providing a clear path to understanding its convergence or divergence.
Formula of Limit Comparison Calculator
The Limit Comparison Test is crucial for studying series. Here is a straightforward explanation of how it works:
- Identify the given series: Let sum a_n be the series you are given.
- Choose a comparison series: Select a series sum b_n whose convergence or divergence is known.
- Compute the limit: Calculate the limit of the ratio of the terms of the two series as n approaches infinity, denoted as (a_n / b_n) = L.
- Analyze the limit:
- If 0 < L < infinity, both series sum a_n and sum b_n either converge or diverge together.
- If L = 0 and sum b_n converges, then sum a_n also converges.
- If L = infinity and sum b_n diverges, then sum a_n also diverges.
Table of Common Series Terms and Their Limits
To aid in the use of the Limit Comparison Calculator, here is a table featuring common terms from various series and their limits:
| Series Term | Comparison Series | Condition for Convergence | Limit Result (L) | Behavior of Series (a_n) |
|---|---|---|---|---|
| 1/n^p | 1/n^2 | p > 1 | Depends on p | Converges if p > 1 |
| 1/(n log n) | 1/n | – | 0 | Converges |
| 1/sqrt(n) | 1/n | – | 0 | Converges |
| n^(-1/2) | n^(-1) | – | 0 | Diverges |
| n^2 | n | – | Infinity | Diverges |
This table serves as a quick reference for users to apply without needing to perform calculations each time.
Example of Limit Comparison Calculator
Let’s demonstrate the use of the Limit Comparison Calculator with an example:
- Given series: sum a_n = 1/n^2
- Comparison series: sum b_n = 1/n^2 (a known convergent p-series where p = 2)
- Calculation of limit: limit as n approaches infinity of (a_n / b_n) = 1
- Analysis: Since the limit is 1 (0 < 1 < infinity), both series converge.
Most Common FAQs
A1: It is most effective with series where the behavior of the comparison series is well-known, such as p-series or geometric series.
A2: It is highly accurate when the input series are correctly define and the known series’ behavior is accurately chosen.
A3: The calculator works best with series that fit the standard forms typically used in academic settings.