Re: recursive limits
- From: Michael Press <jack@xxxxxxx>
- Date: Tue, 07 Nov 2006 09:20:56 GMT
In article
<30903630.1162665044749.JavaMail.jakarta@xxxxxxxxxxxxxx
orum.org>,
romero <void@xxxxxxxxx> wrote:
Let a_(n+2) = (a_(n+1) + a_n)/2 for n >= 1. Prove a_n --> (a_1 + 2*a_2)/3.
Well I tried doing the epsilon delta definition for sequence, but somehow I need to "unroll" the recursive sequence to get down to a_1 and a_2.
Consider linear combinations of the sequences
{c_n} = {1, 1, 1, 1, ...}
{d_n} = {-2, 1, -1/2, 1/4, -1/8, ...}
{a_n} = x * {c_n} + y * {d_n}.
Then x = (1/3) * (a_1 + 2 * a_2), y = (1/3) * (-a_1 + a_2).
Now a_n --> x.
--
Michael Press
.
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