Assume $a$ is the rate of convergence of an algorithm, then how to understand $\frac{1}{1-a}$. Based on the definition, $$ a = \lim_{n \to \infty} \frac{|x_{n+1}-x|}{|x_{n}-x|^{p}}, $$ then if we take $p=1$ (linear convergence), we have $$ \frac{1}{1-a} = \lim_{n \to \infty} \frac{|x_{n}-x|}{|x_{n}-x|-|x_{n+1}-x|} = \lim_{n \to \infty} \frac{e_{n}}{e_{n}-e_{n+1}}, $$ how could we understand it? Thank you very much!
Assume $a$ is the rate of convergence of an algorithm, then how to understand $\frac{1}{1-a}$
3
$\begingroup$
real-analysis
sequences-and-series
convergence
-
0I am wondering where does this question come from, in what context did you become interested in $\dfrac{1}{1-a}$? – 2017-01-30
1 Answers
0
The expression comes from the geometric series $$1+a+a^2+a^3+\cdots=\frac{1}{1-a}$$
I do not know how to prove the given limit, but I think the left side of this idendity is helpful.