We can show that by computing .
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.
Yes, i actually did that, then i think the following can be asserted:
s_{n} converges if and only if s_{n+1} - s_{n} = (x-1)*s_{n} + b_{n+1} converges to 0. I run many MATLAB codes those implying the sequence indeed converges but i couldn't show it rigoruously.