How to Use the Nth Term Test for Divergence

Nth Term Test for Divergence

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.

Infinite Series

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

Convergence

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

Divergence

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

General Term (an)

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

Limit as n approaches infinity

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

Necessary condition for convergence

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

Sufficient condition for convergence

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

Inconclusive Test Result

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

Comparison Test

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

Ratio Test

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

Oscillatory Terms

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

Slowly Decreasing Terms

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

Limit Laws

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

Algebraic Techniques

Mathematical methods used to simplify expressions or calculate limits.

Divergence of a Series

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

Testing for Divergence

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

Power Series

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

Geometric Series

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

Harmonic Series

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

Conditionally Convergent

A series that converges but does not converge absolutely.

Absolutely Convergent

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

Finite Value

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

Step in Series Analysis

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