# Math Help - iterating weighted serie to convergence

1. ## iterating weighted serie to convergence

I have basic weighted function

a = a*(1- alpha) + b*alpha
a = a + alpha*( b - a )

where
0 < alpha < 1

I should figure out how many ( n ) times I have to iterate before
abs( a_n - b) is almost zero.

I have calculated formulas for four iterations if it helps to solve the problem:

a1 = a + alpha*(b-a)
a2 = a + 2*alpha*(b-a) - 1*alpha^2*(b-a)
a3 = a + 3*alpha*(b-a) - 3*alpha^2*(b-a) + alpha^3*(b-a)
a4 = a + 4*alpha*(b-a) - 6*alpha^2*(b-a) + 4*alpha^3*(b-a) - alpha^4*(b-a)

thank you,
Markus

2. Hi

You can show that $a_n-b = (a_0-b)(1-\alpha)^n$

Therefore to get $|a_n-b| < \epsilon$ for some given $\epsilon > 0$ you must have $n \geq \frac{\ln \frac{\epsilon}{|a_0-b|}}{\ln(1-\alpha)}$

3. Great, thanks!