This web page discusses a further test for the convergence or non-convergence of infinte series of positive terms, the ratio test. In some ways it is not as powerful as the comparison test, but the ratio test is particularly useful to test convergence for power series as we shall see.
We are going to look at the ratio test here.
Essentially, the ratio test is just a version of the comparison
test when the sequence to be compared against is
Like the comparison test, the ratio test discusses series of positive terms. This allows us to make some simplifications, via monotonicity for example, as we have already seen.
The Ratio Test.
Then for all
We now look at the two cases when
Apart from what the ratio test actually says, the most important
thing to notice about the ratio test is that it is silent
(i.e., say nothing about convergence or divergence) if
The ratio test generally works well and is the first test to try
for power series such as
The next two examples show that the ratio test really doesn't
say anything when
In fact the series diverges; to prove this use the comparison test
You have seen the ratio test and how it is used to test convergence for series of positive terms. It does not always give an answer. (When it doesn't you should try the comparison test instead.) But when it does work it tends to be easier to use. The ratio test is used particularly for power series.