is a geometric series
if and only if
for some fixed and .
Here we call the common ratio. Notice
that the ratio of any two successive terms is :
Moreover, we have
) if and only if (and, of course
it diverges if ).
We will only prove 1.
Assume that we have
, and notice
, there is
an such that for all we have
Then we have
Here the common ratio for the second one is
thus the right-hand series converges, so the left-hand
The ratio of successive terms is
Thus this series converges absolutely
Note, the minus sign is missing above since in the ratio test
we take the limit of the absolute values.
Thus our series diverges. (Note here that we use that
Let's apply the ratio test to
This tells us nothing.
If this happens... do something else! E.g., in this case, use the