How to Use the Nth Term Test for Divergence

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24 Terms

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Nth Term Test for Divergence

A test used to determine whether an infinite series diverges by examining the behavior of the general term as n approaches infinity.

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Infinite Series

A sum of an infinite sequence of terms, which can either converge to a finite value or diverge.

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Convergence

A property of an infinite series where the sum approaches a finite value.

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Divergence

A property of an infinite series where the sum grows indefinitely or oscillates without settling to a finite value.

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General Term (an)

The n-th term of a series, used in tests to analyze the behavior of the series.

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Limit as n approaches infinity

A mathematical concept indicating the value that a function approaches as the input approaches a certain point, often infinity.

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Necessary condition for convergence

A requirement for a series to converge, such as the individual terms must approach zero.

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Sufficient condition for convergence

A condition that, if satisfied, guarantees the convergence of a series.

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Inconclusive Test Result

A result of the Nth Term Test for Divergence where the limit is zero; further tests are needed.

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Comparison Test

A method to determine the convergence of a series by comparing it to another series.

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Ratio Test

A test for convergence that analyzes the ratio of successive terms in a series.

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Oscillatory Terms

Terms in a series that do not settle towards a single value, often leading to divergence.

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Slowly Decreasing Terms

Terms that approach zero very slowly, which may lead to divergence despite the limit being zero.

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Limit Laws

Rules that govern the calculation of limits, helping to find the limit of functions as they approach a particular point.

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Algebraic Techniques

Mathematical methods used to simplify expressions or calculate limits.

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Divergence of a Series

The process in which the sum of an infinite series does not approach a finite value.

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Testing for Divergence

The process of applying the Nth Term Test to establish whether a series diverges.

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Power Series

An infinite series in the form of a_0 + a_1(x) + a_2(x^2) + ... used to represent functions.

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Geometric Series

A series where each term is a constant ratio multiplied by the previous one.

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Harmonic Series

A specific type of divergent series formed by the sum 1 + 1/2 + 1/3 + 1/4 + ...

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Conditionally Convergent

A series that converges but does not converge absolutely.

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Absolutely Convergent

A series that converges even when the absolute values of the terms are taken.

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Finite Value

A specific numerical value that a series converges to, rather than growing indefinitely.

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Step in Series Analysis

The Nth Term Test serves as an important initial step in evaluating the behavior of an infinite series.