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
- [maths rendering]
- [maths rendering]
Solution
-
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]
-
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.