Series convergence representation

1. Nov 15, 2007

l46kok

1. The problem statement, all variables and given/known data
$$\sum_{n=0}^\infty (0.5)^n * e^{-jn}$$

converges into

$$\frac{1}{1-0.5e^{-jn}}$$

Prove the convergence.

2. Relevant equations

Power series, and perhaps taylor & Macclaurin representation of series.

3. The attempt at a solution

This isn't a homework problem, actually. I just saw this series on the poster and wondered why this is the case (I haven't done series for almost 2 years).

I know for sure that the series has to converge since the $$0.5^n$$ term approaches 0 as n goes to infinity, but I don't understand how the series written above converges into $$\frac{1}{1-0.5e^{-jn}}$$. Can anyone explain?

2. Nov 15, 2007

CompuChip

But isn't this just an ordinary geometric series?
$$\sum_{n = 0}^\infty x^n = \frac{1}{1 - x}$$

3. Nov 15, 2007

l46kok

That's what I was thinking, except that the series is multiplied by an exponential term (with n). And sorry, there was a mistake - there shouldn't be n in the final answer.

4. Nov 15, 2007

CompuChip

Don't get confused over a rewriting of something you already knew
If I'd write it as
$$\sum_{n = 0}^\infty \left( \tfrac12 e^{-j} \right)^n,$$
which is obviously possible since $(e^a)^b = e^{ab}$, would you see it's the same?

5. Nov 15, 2007