3 The ratio test

This test, which is one of the most useful and widely used convergence tests, applies only to series of positive terms.

Key Point 7

The Ratio Test

Let [maths rendering] be a series of positive terms such that, as [maths rendering] increases, the limit of [maths rendering] equals

a number [maths rendering] . That is [maths rendering] .

It can be shown that:

[maths rendering] if [maths rendering] then [maths rendering] diverges

[maths rendering] if [maths rendering] then [maths rendering] converges

[maths rendering] if [maths rendering] then [maths rendering] may converge or diverge.

That is, the test is inconclusive in this case.

Example 1

Use the ratio test to examine the convergence of the series

  1. [maths rendering]
  2. [maths rendering]
Solution
  1. The general term in this series is [maths rendering] i.e.

    [maths rendering]

    and the ratio

    [maths rendering]

    [maths rendering]

    Since [maths rendering] the series is convergent. In fact, it will be easily shown, using the techniques outlined in HELM booklet  16.5, that

    [maths rendering]

  2. Here we must assume that [maths rendering] since we can only apply the ratio test to a series of positive terms.

    Now

    [maths rendering]

    so that

    [maths rendering]

    and

    [maths rendering]

    Thus, using the ratio test we deduce that (if [maths rendering] is a positive number) this series will only converge if [maths rendering]

    We will see in Section 16.4 that

    [maths rendering]

Task!

Use the ratio test to examine the convergence of the series:

[maths rendering]

First, find the general term of the series:

[maths rendering]

Now find [maths rendering] :

[maths rendering]

Finally, obtain [maths rendering] :

[maths rendering] . Now [maths rendering] as [maths rendering] increases

[maths rendering]

Hence this is a convergent series.

Note that in all of these Examples and Tasks we have decided upon the convergence or divergence of various series; we have not been able to use the tests to discover what actual number the convergent series converges to.