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

Ratio Test for Convergence (College Board AP® Calculus BC)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Mark Curtis

Updated on

Ratio test

What is the ratio test?

  • The ratio test is a method for determining whether an infinite series converges or diverges

  • Let sum from n equals 1 to infinity of a subscript n be a series

    • If limit as n rightwards arrow infinity of open vertical bar a subscript n plus 1 end subscript over a subscript n close vertical bar less than 1, the series converges

    • If limit as n rightwards arrow infinity of open vertical bar a subscript n plus 1 end subscript over a subscript n close vertical bar greater than 1 or if the limit is infinite, the series diverges

    • If limit as n rightwards arrow infinity of open vertical bar a subscript n plus 1 end subscript over a subscript n close vertical bar equals 1, the ratio test gives no information about convergence

      • In this case another test must be used to determine whether the series converges or diverges

  • Note that if all the terms of the series are positive (i.e. if a subscript n greater than 0 for all values of n)

    • then you can consider the limit limit as n rightwards arrow infinity of a subscript n plus 1 end subscript over a subscript n instead (i.e., without needing the absolute value sign)

Examiner Tips and Tricks

Use the ratio test for series whose terms include more complicated expressions like exponentials, logarithms and factorials. The ratio test will generally be inconclusive for series with terms that are rational functions with only polynomials in the numerator and/or denominator.

Worked Example

Apply the ratio test to each of the following series.

(a) sum from n equals 1 to infinity of fraction numerator n factorial over denominator e to the power of n end fraction

All the terms here are positive, so we won't need the absolute value sign when taking the limit

Write and simplify the ratio a subscript n plus 1 end subscript over a subscript n

table row cell fraction numerator fraction numerator open parentheses n plus 1 close parentheses factorial over denominator e to the power of n plus 1 end exponent end fraction over denominator fraction numerator n factorial over denominator e to the power of n end fraction end fraction end cell equals cell fraction numerator open parentheses n plus 1 close parentheses factorial over denominator e to the power of n plus 1 end exponent end fraction times fraction numerator e to the power of n over denominator n factorial end fraction end cell row blank equals cell fraction numerator open parentheses n plus 1 close parentheses times n factorial times e to the power of n over denominator e times e to the power of n times n factorial end fraction end cell row blank equals cell fraction numerator n plus 1 over denominator e end fraction end cell end table

Take the limit; remember that e equals 2.718281... is just a constant

limit as n rightwards arrow infinity of a subscript n plus 1 end subscript over a subscript n equals limit as n rightwards arrow infinity of fraction numerator n plus 1 over denominator e end fraction equals infinity greater than 1

The limit is greater than 1 (it diverges to positive infinity), so the series is divergent

According to the ratio test the series diverges

(b)sum from n equals 1 to infinity of open parentheses negative 1 close parentheses to the power of n open parentheses ln 2 close parentheses to the power of n over n to the power of 4

Write and simplify the ratio a subscript n plus 1 end subscript over a subscript n

table row cell fraction numerator open parentheses negative 1 close parentheses to the power of n plus 1 end exponent open parentheses ln 2 close parentheses to the power of n plus 1 end exponent over open parentheses n plus 1 close parentheses to the power of 4 over denominator open parentheses negative 1 close parentheses to the power of n open parentheses ln 2 close parentheses to the power of n over n to the power of 4 end fraction end cell equals cell fraction numerator open parentheses negative 1 close parentheses to the power of n plus 1 end exponent times open parentheses ln 2 close parentheses to the power of n plus 1 end exponent over denominator open parentheses n plus 1 close parentheses to the power of 4 end fraction times fraction numerator n to the power of 4 over denominator open parentheses negative 1 close parentheses to the power of n times open parentheses ln 2 close parentheses to the power of n end fraction end cell row blank equals cell open parentheses negative 1 close parentheses to the power of n plus 1 end exponent over open parentheses negative 1 close parentheses to the power of n times fraction numerator ln 2 times open parentheses ln 2 close parentheses to the power of n times n to the power of 4 over denominator open parentheses ln 2 close parentheses to the power of n times open parentheses n plus 1 close parentheses to the power of 4 end fraction end cell row blank equals cell open parentheses negative 1 close parentheses times ln 2 times open parentheses fraction numerator n over denominator n plus 1 end fraction close parentheses to the power of 4 end cell row blank equals cell negative ln 2 times open parentheses fraction numerator n over denominator n plus 1 end fraction close parentheses to the power of 4 end cell row blank equals cell negative ln 2 times open parentheses fraction numerator n over denominator n plus 1 end fraction times fraction numerator bevelled 1 over n over denominator bevelled 1 over n end fraction close parentheses to the power of 4 end cell row blank equals cell negative ln 2 times open parentheses fraction numerator 1 over denominator 1 plus 1 over n end fraction close parentheses to the power of 4 end cell end table

Take the limit

table row cell limit as n rightwards arrow infinity of open vertical bar a subscript n plus 1 end subscript over a subscript n close vertical bar end cell equals cell limit as n rightwards arrow infinity of open vertical bar negative ln 2 times open parentheses fraction numerator 1 over denominator 1 plus 1 over n end fraction close parentheses to the power of 4 close vertical bar end cell row blank equals cell limit as n rightwards arrow infinity of ln 2 times open parentheses fraction numerator 1 over denominator 1 plus 1 over n end fraction close parentheses to the power of 4 end cell row blank equals cell ln 2 times open parentheses fraction numerator 1 over denominator 1 plus 0 end fraction close parentheses to the power of 4 end cell row blank equals cell ln 2 end cell row blank equals cell 0.693147... less than 1 end cell end table

The limit is less than 1, so the series converges

Note that if ln 2 in the series term formula were changed to ln 3 equals 1.098612..., then the series would diverge!

According to the ratio test the series converges

(c) sum from n equals 1 to infinity of 1 over n squared

This is a p-series with n equals 2, so we already know it converges, but apply the ratio test nonetheless

All the terms here are positive, so we won't need the absolute value sign when taking the limit

Write and simplify the ratio a subscript n plus 1 end subscript over a subscript n

table row cell fraction numerator 1 over open parentheses n plus 1 close parentheses squared over denominator 1 over n squared end fraction end cell equals cell 1 over open parentheses n plus 1 close parentheses squared times n squared over 1 end cell row blank equals cell n squared over open parentheses n plus 1 close parentheses squared end cell row blank equals cell open parentheses fraction numerator n over denominator n plus 1 end fraction close parentheses squared end cell row blank equals cell open parentheses fraction numerator n over denominator n plus 1 end fraction times fraction numerator bevelled 1 over n over denominator bevelled 1 over n end fraction close parentheses squared end cell row blank equals cell open parentheses fraction numerator 1 over denominator 1 plus 1 over n end fraction close parentheses squared end cell end table

Take the limit

table row cell limit as n rightwards arrow infinity of a subscript n plus 1 end subscript over a subscript n end cell equals cell limit as n rightwards arrow infinity of open parentheses fraction numerator 1 over denominator 1 plus 1 over n end fraction close parentheses squared end cell row blank equals cell open parentheses fraction numerator 1 over denominator 1 plus 0 end fraction close parentheses squared end cell row blank equals 1 end table

The limit is equal to 1, so the test is inconclusive

Note that the ratio test would similarly fail for the divergent harmonic series sum from n equals 1 to infinity of 1 over n

The ratio test cannot determine whether this series converges or diverges

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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.