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AP®MathsCollege BoardCalculus BCStudy GuidesUnit 10: Infinite Sequences and SeriesIntroduction to Infinite SeriesHarmonic Series & p-series

Harmonic Series & p-series (College Board AP® Calculus BC)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Mark Curtis

Updated on

Harmonic series

What is the harmonic series?

  • The harmonic series is the series sum from n equals 1 to infinity of 1 over n equals 1 plus 1 half plus 1 third plus 1 fourth plus...

  • The harmonic series diverges

Alternating harmonic series

What is the alternating harmonic series?

  • The alternating harmonic series is the 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...

  • The alternating harmonic series converges

  • The sum of the alternating harmonic series is 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 ln space 2

    • I.e. the sum converges to the natural logarithm of 2

p-series

What are p-series?

  • A p-series is a series of the form sum from n equals 1 to infinity of 1 over n to the power of p equals 1 plus 1 over 2 to the power of p plus 1 over 3 to the power of p plus 1 over 4 to the power of p plus...

    • where p is a constant power

    • Note that p does not need to be an integer

  • A p-series converges if p greater than 1

  • A p-series diverges if p less or equal than 1

  • Note that the harmonic series is a p-series with p equals 1

Examiner Tips and Tricks

Make sure you are familiar with the harmonic series, alternating harmonic series, and p-series, along with their respective convergence results. These results will often be needed to prove the convergence or divergence of other series.

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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.