Geometric Series
Why
It is believable that has a convergent series.
And likewise with
What of for .
Definition
Let .
The geometric series of
is the series of the sequence
defined by .
Characterization of convergence
Does the geometric series of converge?
In other words, does defined by have a limit.
For and , we have seen (see Real Series) that the series diverges.
However for the cases and
the geometric series converges.
If , then the geometric series
of converges and
If then the geometric series
of diverges.
Define .
Then
From which we deduce, .
If , then
If , then using the algebra
of limits (see Real Limit Algebra) we deduce
since for
.
If or , then we have seen
that diverges.