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AP®MathsCollege BoardCalculus BCStudy GuidesUnit 10: Infinite Sequences and SeriesTests for Divergence & ConvergenceAlternating Series Test for Convergence

Alternating Series Test for Convergence (College Board AP® Calculus BC)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Mark Curtis

Updated on

Alternating series test

What is an alternating series?

  • An alternating series is a series where the terms alternate between negative and positive

    • For example the alternating harmonic series sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent over n equals 1 minus 1 half plus 1 third minus 1 fourth plus...

  • Alternating series can be written in one of the following two forms, where in each case a subscript n greater than 0 for each value of n

    • sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent times a subscript n equals a subscript 1 minus a subscript 2 plus a subscript 3 minus a subscript 4 plus...

      • This version starts with a positive term

    • sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n times a subscript n equals negative a subscript 1 plus a subscript 2 minus a subscript 3 plus a subscript 4 minus...

      • This version starts with a negative term

What is the alternating series test for convergence?

  • The alternating series test is a method for determining whether an alternating series converges

  • For an alternating series of the form sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent times a subscript n or sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n times a subscript n with a subscript n greater than 0 for each value of n, the series converges if

    • a subscript 1 greater or equal than a subscript 2 greater or equal than a subscript 3 greater or equal than... greater or equal than a subscript n greater or equal than...

    • and limit as n rightwards arrow infinity of a subscript n equals 0

  • Note that you can't use the alternating series test to show that an alternating series diverges

    • There are convergent alternating series for which a subscript 1 greater or equal than a subscript 2 greater or equal than a subscript 3 greater or equal than... greater or equal than a subscript n greater or equal than... is not true

    • However limit as n rightwards arrow infinity of a subscript n equals 0 must be true for any series to converge (see the 'nth Term Test for Divergence' study guide)

Worked Example

Use the alternating series test to show that the alternating harmonic series sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent over n equals 1 minus 1 half plus 1 third minus 1 fourth plus... converges.

The harmonic series can be rewritten as sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent times 1 over n, i.e. in 'standard' alternating series form with a subscript n equals 1 over n

First show that a subscript 1 greater or equal than a subscript 2 greater or equal than a subscript 3 greater or equal than... greater or equal than a subscript n greater or equal than... is true

1 greater than 1 half greater than 1 third greater than 1 fourth greater than... greater than 1 over n greater than...

Now check the limit as n rightwards arrow infinity of a subscript n equals 0 condition

limit as n rightwards arrow infinity of 1 over n equals 0

Therefore both conditions of the alternating series test have been met

By the alternating series test, sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n plus 1 end exponent over n converges

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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.