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

**fardeen_gen** · April 30th 2009, 08:23 PM

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

**Showcase_22** · May 1st 2009, 01:22 AM

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

$<br /> <br /> 1 + 2\cos^2 \theta_{1}Z + 2\cos^2\theta_{2}Z^2 + ... + 2\cos^2\theta_{n}Z^n = 0<br />$

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

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

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

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

I'm not really sure where to go next

**CaptainBlack** · May 1st 2009, 02:34 AM

Originally Posted by fardeen_gen

If $|Z| < \frac{1}{3}$ , then prove that the equation $1 + 2\cos^2 \theta_{1}Z + 2\cos^2\theta_{2}Z^2 + ... + 2\cos^2\theta_{n}Z^n = 0$ , 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

**Showcase_22** · May 1st 2009, 03:09 AM

aww, shoot!

I was way off!!

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

LinkBack

Thread Tools

Display

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

Similar Math Help Forum Discussions

Prove that complex number can represented in this form

partial differential equation satisfied by u(x,y)

[SOLVED] Prove that an essential singularity approaches every complex number.

complex number equation

Prove that there are always two square roots of a non zero complex number

Search Tags