Thread: Convergence rate

I have the following sequences of numbers:
1) 32, 16, 8, 4, 2, 1
2) 32, 16, 8, 2, 1, 0.5
3) 32, 16, 4, 2, 1, 0.5,
4) 32, 16, 4, 1, 0.25, 0.0625

I'm interested which of the above has the fastest rate of convergence.
(mathematical formula is needed). I should be able to plot the rates of
convergence against each other, and it should be concluded from the plot.
(The data given above are just the illustration)

Originally Posted by onako

I have the following sequences of numbers:
1) 32, 16, 8, 4, 2, 1
2) 32, 16, 8, 2, 1, 0.5
3) 32, 16, 4, 2, 1, 0.5,
4) 32, 16, 4, 1, 0.25, 0.0625

I'm interested which of the above has the fastest rate of convergence.
(mathematical formula is needed). I should be able to plot the rates of
convergence against each other, and it should be concluded from the plot.
(The data given above are just the illustration)

If you suppose that each sequence is of the form $\displaystyle a(i)=A i^{-k}$ , then the larger $\displaystyle $$ k$ is the faster the sequence goes to zero.

Take logs and you get:

$\displaystyle \log(a(i))=-k\log(i) +\log(A)$

so put $\displaystyle y=\log(a(i))$ and $\displaystyle x=\log(i)$ and plot, and the slopes of the graphs for the sequences indicate their rate of convergence.

CB