Math Is Fun Forum

MAD

Member

Registered: 2011-03-15

Posts: 2

convergence of a sequence

Hi,

I have a problem in my mind, it is as follows:

We know that the sum of positive integer powers of a number x s.t. x is between 0 and 1 converges to 1/(1-x). Furthermore, assume that we have sequence b_{n} converging to b>0. Let

s_{n} = {sum}_{i=0}^{n}[x^{n-i}*b_{i}]     (in words: s_{n} equals to sum from i=0 to n of the terms x to the n-i multiplied by b_{i})

The question is, does s_{n} converges?{may be helpful: if it converges, it converges to b/(1-x)}.

Thanks.

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: convergence of a sequence

try this
consider two vectors 

By Schwarz's inequality, 

converges

We know C_n is convergent, C_n is bounded by some number M
for any epsilon > 0,  there exist  N, such that |C_n - b| < epsilon/ M  for n>N
then

So

converges as well
Hence

converges too, since its increasing and non negative.