1. ## Another summation problem_

If we have q= Σ exp(2*pi*j*k*n/N) , for n=0 to M -1

I want to prove a condition for M such as q=0 . .

I'm a bit confused. I start to solve it by knowing that

Σ exp(2*pi*j*k*n/N) , for n=0 to N-1 , = 1- exp(2pi*j*k)/ 1-exp(2pi*j*k/N)

...

Nikolas

2. Originally Posted by tsebamm
If we have q= Σ exp(2*pi*j*k*n/N) , for n=0 to M -1

I want to prove a condition for M such as q=0 . .

I'm a bit confused. I start to solve it by knowing that

Σ exp(2*pi*j*k*n/N) , for n=0 to N-1 , = 1- exp(2pi*j*k)/ 1-exp(2pi*j*k/N)

...

Nikolas
Sometimes you write N, sometimes M...a mess. Anyway, you already have it!:

$\displaystyle{\frac{1-e^{2\pi jk}}{1-e^{2\pi jk/N}}=0$ , of course assuming $j,k\in\mathbb{Z}\,,\,jk\neq 0\!\!\pmod N$ , so $M=N$ does the trick.

Tonio

3. Originally Posted by tonio
Sometimes you write N, sometimes M...a mess. Anyway, you already have it!:

$\displaystyle{\frac{1-e^{2\pi jk}}{1-e^{2\pi jk/N}}=0$ , of course assuming $j,k\in\mathbb{Z}\,,\,jk\neq 0\!\!\pmod N$ , so $M=N$ does the trick.

Tonio
Thanks for that. But I do not think that I understand it. When M=N then q=0 ?? What's the proof of that?