The Ratio Test
A power series is an expression of the form
where 
and the 
are constants. Power series arise
in a number of contexts; the Taylor series of a function is the most
prominent in calculus. A power series such as (1)
defines a function for
all values of x where this sum converges. We would
like to determine these values of x (i.e., the domain of
the power series). The ratio test provides a quick method
of determining almost all of these values.
Here is the statement:
The Ratio Test
Given a series 
where 
for n sufficiently large,
,
then the sum converges;
,
then the sum diverges.
Remark:
if 
,
the theorem makes no claims at all; the test is not conclusive in this
case and one would have to use other means to determine whether or not
the sum converges.
Remark: If 
, then the ratio
test applies implying that the series diverges.
The idea
behind the ratio test is that if 
, then for n large each
. In other words, the series will
behave like a geometric series with ratio r. A geometric
series converges iff its ratio r satisfies |r|<1.
