Intervals of convergence of power series

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• February 26th 2006, 02:01 PM
bsteel
Intervals of convergence of power series
I need some homework help for finding the interval of convergence for the series (x/2)^n where n starts at n=0 and ends at n=infinity.
• February 26th 2006, 10:33 PM
CaptainBlack
Quote:

Originally Posted by bsteel
I need some homework help for finding the interval of convergence for the series (x/2)^n where n starts at n=0 and ends at n=infinity.

The series you are interested in the convergence of may be written:

$
\sum_{n=0}^{\infty} (x/2)^n=\lim_{N\rightarrow \infty}\sum_{n=0}^{N} (x/2)^n
$

Now:

$
\left(\sum_{n=0}^{N} (x/2)^n \right)(1-(x/2))=1-(x/2)^{N+1}
$

(If you multiply out the product on the Left Hand Side you will find
all but the first and last terms cancel).

So:

$
\sum_{n=0}^{N} (x/2)^n=\frac{1-(x/2)^{N+1}}{1-(x/2)}
$

Hence

$
\lim_{N\rightarrow \infty}\sum_{n=0}^{N} (x/2)^n
$

converges if and only if:

$
\lim_{N\rightarrow \infty} \frac{1-(x/2)^{N+1}}{1-(x/2)}
$

converges, and this last expression converges if and only if

$
\left| \frac{x}{2}\right| <1
$

or, if and only if:

$
-2$

RonL