Example 6.2.2 (Geometric series)
Consider the
geometric series
![$ \sum_{n=1}^{\infty} a r^{n-1}$](img978.png)
for
![$ a\neq 0$](img979.png)
.
Then
To see this, multiply both sides by
![$ 1-r$](img981.png)
and notice
that all the terms in the middle cancel out.
For what values of
![$ r$](img230.png)
does
![$ \lim_{N\to\infty} \frac{a(1-r^N)}{1-r}$](img982.png)
converge?
If
![$ \vert r\vert<1$](img983.png)
, then
![$ \lim_{N\to\infty} r^N = 0$](img984.png)
and
If
![$ \vert r\vert> 1$](img986.png)
, then
![$ \lim_{N\to\infty} r^N$](img987.png)
diverges,
so
![$ \sum_{n=1}^{\infty} a r^{n-1}$](img978.png)
diverges.
If
![$ r=\pm 1$](img988.png)
, it's clear since
![$ a\neq 0$](img979.png)
that the
series also diverges (since the partial sums are
![$ s_N = \pm Na$](img989.png)
).
For example, if
and
, we get
as claimed earlier.