Nth Term Test for Divergence

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33 Terms

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Nth Term Test

A method used to check if an infinite series diverges by examining the general term as n approaches infinity.

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General Term

The expression a_n in an infinite series that represents the n-th term of the series.

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Convergence Condition

For a series to converge, its general term a_n must approach 0 as n approaches infinity.

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Divergence Condition

If the limit of the general term a_n does not approach 0, the series cannot converge.

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Limit Notation

lim_{n o ext{∞}} a_n is used to denote the limit of the general term as n approaches infinity.

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Series Analysis

The process of evaluating the behavior of an infinite series to determine convergence or divergence.

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Necessary Condition for Convergence

The general term a_n must approach 0 for the series to have the potential to converge.

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Insufficient Condition for Convergence

A limit of 0 for a_n does not guarantee convergence; further testing is required.

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Further Tests

Tests like the Comparison Test, Ratio Test, or Integral Test used when lim_{n o ∞} a_n = 0.

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Example of Divergence

A scenario where the limit of the general term does not approach 0, leading to series divergence.

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Evaluation Techniques

Methods involving algebra or calculus to compute the limit of the general term.

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Behavior of Series

Refers to how the terms of an infinite series interact to determine convergence or divergence.

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Limit Evaluation

The process of finding the value that the general term a_n approaches as n becomes very large.

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Comparison Test

A method used to compare the behavior of two series to determine convergence or divergence.

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Ratio Test

A method that examines the lim_{n o ∞} |a_{n+1}/a_n| to determine convergence or divergence.

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Integral Test

A method that uses integrals to assess the convergence of a series.

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Sum of Infinite Terms

The result of adding infinitely many terms together, which may or may not yield a finite value.

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Limit Zero Condition

The condition in which lim_{n o ∞} a_n = 0, indicating that the series requires further tests.

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Quick First Step

The Nth Term Test as an efficient initial approach to assess series divergence.

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Simplified Analysis

A simplified way to look at series behavior without deep complex methods.

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Key Limitation

The Nth Term Test cannot confirm convergence, only indicate potential divergence.

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Conclusion of the Test

The Nth Term Test aids in identifying divergent series but necessitates additional tests for comprehensive analysis.

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Series Behavior Understanding

The need for multiple testing methods to fully analyze how a series converges or diverges.

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Inherently Divergent Series

A series that, by its term structure, showcases divergence when the general term does not approach zero.

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Limit of General Term

The value that the general term approaches, essential for determining convergence.

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Application Steps

Identify the general term, compute the limit, and interpret the result.

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Divergence Test

Another name for the Nth Term Test, focusing on confirming divergence of a series.

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Interpretation of Results

The conclusion drawn based on the limit value of a_n as n approaches infinity.

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Behavioral Insight

The understanding gained from observing how a series behaves under the limit condition.

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Initial Divergence Indicator

A non-zero limit from the Nth Term Test suggests immediate divergence indication.

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Long-term Series Behavior

The overall tendency of a series when subjected to infinite conditions.

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Mathematical Justification

The theoretical underpinning for why the Nth Term Test is effective for divergence.

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Proof of Divergence

Demonstrating that a series is not converging through Nth Term Test findings.