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Limit Comparison Calculator Online

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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.
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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 TermComparison SeriesCondition for ConvergenceLimit Result (L)Behavior of Series (a_n)
1/n^p1/n^2p > 1Depends on pConverges if p > 1
1/(n log n)1/n0Converges
1/sqrt(n)1/n0Converges
n^(-1/2)n^(-1)0Diverges
n^2nInfinityDiverges

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.
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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.

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