Intervals of convergence of power series

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<x<2 $

RonL