Proving the Generating Function for 1/(z-1) Using Binomial Expansion

[nowimpsbball]

Homework Statement

Prove that the generating function 1/(1-z) = (1+z)(1+z^2)(1+z^4)(1+z^8)...
which is also to 1+z+z^2+z^3+z^4+... when you multiply out the binomials.

Homework Equations

(1/(1-z))^k = {$$\Sigma$$[from i=0 to infinity] C(i+k-1, k-1)z^i}

The Attempt at a Solution

I've been playing around with this for a while. I thought I had it, but then I realized that I plugged the "n+1" into the exponent...ie 1+z+z^2+z^3+z^4+...+z^n+z^n+1...but that is not right...I need to plug in z+1 into each z.

PS Ignore the topic name, it should be 1/(1-z)

Thanks
Thanks

 

[e(ho0n3]

So you want to prove that 1/(1-z) = (1+z)(1+z²)(1+z⁴)...?

 

[HallsofIvy]

Why do you refer to induction? There is no "n" in your formula to do an induction on!