How Do You Prove the Sum of a Geometric Series in Complex Numbers?

[cepheid]

Prove:

$$\sum^{n}_{k=0} {z^{k}} = \frac{z^{n+1} -1}{z-1}$$

$$z \in \mathbb{C}$$

I have no idea where to even start. Just some hints on the strategy/pattern we are supposed to see would be great.

 

[cepheid]

Oh yeah, I forgot:

$$k \in \mathbb{N}$$

 

[dextercioby]

HINT:Consider the polynomials
$$P_{1}(z)=z^{n}-1$$
$$P_{2}(z)=z-1$$

Can u show that $¨P_{2}/P_{1}$??.Then simply divide the first through the second...

Daniel.

EDIT:Then make n->n+1 and u'll get your result...

 

[quasar987]

Exactly the same way you prove the sum of a finite geometric serie.

The proof I'm talking about is done in mathworld: http://mathworld.wolfram.com/GeometricSeries.html

 

[cepheid]

quasar:

Thanks for the reminder! I had forgotten that derivation. It seems that if you subtract Sn from zSn instead of the other way around like they did, then you'll arrive at the answer I'm supposed to show. Obviously it doesn't matter, since both numerator and denominator would be the negative of what they were if you did it the other way 'round. Thanks.

dexter:

"Can u show that P2/P1 ?? "...is not quite a complete thought. I'm just curious as to what about them you intended for me to show.

Thanks.

 

[dextercioby]

What is the solution of the eq.$P_{2}=0$.Is it unique??If so,then,if [itex]P_{1}(1) = 0[/tex],it means that the second polynomial (P_{2}) divides the first.Then u can divide the first through the second...<br /> <br /> Daniel.[/itex]