Series convergence representation

[asd1249jf]

Homework Statement

$$\sum_{n=0}^\infty (0.5)^n * e^{-jn}$$

converges into

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

Prove the convergence.

Homework Equations

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

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?

 

[CompuChip]

There shouldn't be an n in your final answer, obviously.
But isn't this just an ordinary geometric series?
$$\sum_{n = 0}^\infty x^n = \frac{1}{1 - x}$$

 

[asd1249jf]

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.

 

[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?

 

[asd1249jf]

[Hits Head]

Doh, of course. Thanks