Proof that a Sequence ConvergesDate: 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 Check out our web site! http://mathforum.org/dr.math/ |
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