Proof that a Sequence Converges

Date: 8/23/96 at 4:59:13
From: Anonymous
Subject: Proof: Sequence Converges

Prove that, if | a | < 2 for all i = 1,2,3,..n, then the equation
                  i

   a z + a z^2 + a z^3 + ...+ a z^n = 1
    1     2       3            n

has no roots within | z | < 1/3 .

Date: 8/23/96 at 8:20:3
From: Doctor Anthony
Subject: Re: Proof: Sequence Converges

If we wish |z| to be a minimum then the |a_i| should be a maximum, 
which is 2.  Thus we require:

   2{|z| + |z^2| + |z^3| + ....+...} = 1,

or, applying the triangle inequality in its general form,

   2*|z + z^2 + z^3 + ...| < 1
   2*|z|/(1-|z|) < 1
   2|z| < 1 - |z|
   3|z| < 1      and |z| < 1/3.

-Doctor Anthony,  The Math Forum
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