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AP®MathsCollege BoardCalculus BCStudy GuidesUnit 10: Infinite Sequences and SeriesTests for Divergence & Convergencenth Term Test for Divergence

nth Term Test for Divergence (College Board AP® Calculus BC)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Mark Curtis

Updated on

nth term test

What is the nth term test?

  • The nth term test is a way to show that an infinite series diverges

    • You cannot show that a series converges using this test

  • For a series sum from space n equals 1 to infinity of a subscript n

    • if limit as n rightwards arrow infinity of a subscript n not equal to 0 then the limit diverges

  • Note that if limit as n rightwards arrow infinity of a subscript n equals 0, this does not necessarily mean that the series converges

    • Other tests must be used to determine convergence or divergence of such a series

Examiner Tips and Tricks

The nth term test is a good 'first test' to use when examining a series, as it is usually quick and simple to apply.

Worked Example

Determine whether the series sum from n equals 1 to infinity of fraction numerator 3 n squared over denominator 1 minus 5 n squared end fraction converges or diverges.

Consider the limit of the nth term, limit as n rightwards arrow infinity of fraction numerator 3 n squared over denominator 1 minus 5 n squared end fraction

fraction numerator 3 n squared over denominator 1 minus 5 n squared end fraction equals fraction numerator 3 n squared over denominator 1 minus 5 n squared end fraction times fraction numerator bevelled 1 over n squared over denominator bevelled 1 over n squared end fraction equals fraction numerator 3 over denominator 1 over n squared minus 5 end fraction

limit as n rightwards arrow infinity of open parentheses fraction numerator 3 n squared over denominator 1 minus 5 n squared end fraction close parentheses equals limit as n rightwards arrow infinity of open parentheses fraction numerator 3 over denominator 1 over n squared minus 5 end fraction close parentheses equals fraction numerator 3 over denominator 0 minus 5 end fraction equals negative 3 over 5

That limit is not equal to zero, so the series diverges

limit as n rightwards arrow infinity of fraction numerator 3 n squared over denominator 1 minus 5 n squared end fraction not equal to 0 space so the sequence diverges by the nth term test

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.