# Thread: Prove the equation won't be satisfied for any complex number Z?

1. ## Prove the equation won't be satisfied for any complex number Z?

If $\displaystyle |Z| < \frac{1}{3}$, then prove that the equation $\displaystyle 1 + 2\cos^2 \theta_{1}Z + 2\cos^2\theta_{2}Z^2 + ... + 2\cos^2\theta_{n}Z^n = 0$, will not be satisfied by any complex number z.

2. I haven't solved it yet, but here's what I have so far:

$\displaystyle 1 + 2\cos^2 \theta_{1}Z + 2\cos^2\theta_{2}Z^2 + ... + 2\cos^2\theta_{n}Z^n = 0$

$\displaystyle 1+(1+ \cos 2 \theta_1)Z+(1+ \cos 2 \theta_2)Z^2+....+ (1+\cos 2 \theta_n)Z^n=0$

$\displaystyle (1+Z+Z^2+.......+Z^n)+(Z \cos 2\theta_1+ Z^2 \cos 2\theta_2+.....+ Z^n \cos 2 \theta_n)=0$

$\displaystyle 0=(1+Z+Z^2+.......+Z^n)+(Z \cos \theta_2+ Z^2 \cos \theta_2+.....+ Z^n \cos 2 \theta_n)$$\displaystyle < \left(1+\frac{1}{3}+\frac{1}{9}+......+\frac{1}{3^ n} \right)+ \left( \frac{1}{3} \cos 2\theta_1+\frac{1}{9} \cos 2\theta_2+.....+ \frac{1}{3^n} \cos 2 \theta_n \right)$

$\displaystyle \left( 1+\frac{1}{3}+\frac{1}{3^2}+...+ \frac{1}{3^n} \right)= \frac{1-\left( \frac{1}{3} \right)^n}{1-\frac{1}{3}}=\frac{3}{2} \left( 1- \left( \frac{1}{3} \right)^n \right)$

I'm not really sure where to go next

3. Originally Posted by fardeen_gen
If $\displaystyle |Z| < \frac{1}{3}$, then prove that the equation $\displaystyle 1 + 2\cos^2 \theta_{1}Z + 2\cos^2\theta_{2}Z^2 + ... + 2\cos^2\theta_{n}Z^n = 0$, will not be satisfied by any complex number z.
Because the Cauchy lower bound on the roots of this polynomial is greater than or equal to 1/3.

(see here)

CB

4. aww, shoot!

I was way off!!