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.

Suppose _{
}
_{
+1
}_{
}

(a) if _{
}

(b) if _{
}

**Proof.**

Since each _{
0
}, we have
_{
+1
}_{
}

Let _{0
}

Then for all _{0}

so

We now look at the two cases when

If _{
+1
(-)}_{
}
for all _{0}

so _{
}
^{
}

If _{
+1
(+)}_{
}
for all _{0}

so _{
}
^{
}

Example.

Let ^{
}
_{
=
} Then

as _{
=
} so this sequence (_{
)
} is unbounded
and hence non-convergent (by
boundedness of convergent sequences)
and hence not null. Therefore the series diverges for

The ratio test generally works well and is the first test to try
for power series such as _{
}
_{
} are positive. If some terms
are non-positive, see the section later on absolute convergence for
information on what to do.

The next two examples show that the ratio test *really* doesn't
say anything when

Example.

Consider _{
=11+
}. Then _{
+1
}
_{
}

In fact the series diverges; to prove this use the comparison test
with

Example.

Consider _{
=1(1+)2
}. Then _{
+1
}
_{
}
^{2}
^{2}
^{2}
^{2}

In fact the series converges; to prove this use the comparison test
with ^{2}

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.