if l ak l (k is as a subscript)

Question

if l ak l (k is as a subscript)<3 ,1<k<n then all the complex numbers z satisfy 1+a1*z+a 2*z^2+a3*z^3+.....+an*z^n(1,2,3,....n in a1,a2,...an are in subscript)=0 than lzl lie in the range?

anonymous · Answer

$|a_k|<3,1<k<n$ and consider all $z$ satisfy $0=1+a_1 z+a_2 z^2+a_3z^3+\dots+a_n z^n$... so$$0=1+a_1z+\dots+a_nz^n\0=|1+a_1z+\dots+a_n z^n|\le 1+|a_1||z|+\dots+|a_n||z|^n$$

anonymous · Answer

and $$1+|a_1||z|+\dots+|a_n||z|^n<1+3|z|+\dots+3|z|^n$$but$$1+3|z|+\dots+3|z|^n=1+3|z|(1+\dots+|z|^{n-1})=1+3|z|\cdot\frac{1-|z|^n}{1-|z|}$$

anonymous · Answer

hmm...

anonymous · Answer

http://en.wikipedia.org/wiki/Properties_of_polynomial_roots

anonymous · Answer

http://en.wikipedia.org/wiki/Properties_of_polynomial_roots#Based_on_the_Rouch.C3.A9_theorem can't complete it but this is what they want you to do ^^ sorry I have to get ready for class