## 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

**Q1: Can the Comparison Calculator be use for any series?**A1: It is most effective with series where the behavior of the comparison series is well-known, such as p-series or geometric series.

**Q2: How accurate is the Comparison Calculator?**A2: It is highly accurate when the input series are correctly define and the known series’ behavior is accurately chosen.

**Q3: Is there a limit to the complexity of series the calculator can handle?**A3: The calculator works best with series that fit the standard forms typically used in academic settings.