Rate of Convergence: Definition, Calculus & Examples

[petermer]

Hi to all! I'm new to calculus and would like to know how to find the rate of convergence for a function. I'm aware of the Wikipedia article, but it only defines it for a sequence. So, what is the general definition?

 

[Irrational]

what are you hoping your function converges on?

i would have imagined you are trying to find the rate of convergence of a sequence approximating the function.

 

[petermer]

For example, I'd like to find the rate of convergence of lnx as it approaches infinity.

 

[Irrational]

as you sure you're not trying to find the limit of ln(x) as x approaces infinity?

the limit as x goes to infinity of ln(x) is infinity. what are you hoping ln(x) converges on?

if you're talking about the rate of convergence of the taylor series expansion of ln(x), the series only converges in the range -1 <= x < 1. it diverges outside this range so makes not sense to test the rate of convergence of it as x goes to infinity.

to be honest, i don't understand what you are asking.

 

[petermer]

I do know it's limit, but I'm trying to find the rate (name it 'velocity') with which this function converges to it's limit, infinity. I'm referring to http://en.wikipedia.org/wiki/Rate_of_convergence" Wikipedia article. There, for example, it is mentioned that the sequence 1/2^x converges to it's limit to infinity, 0, with a rate of 1/2. I just look for a more generalized version of this method.

 

[Irrational]

from your wiki link, the speed at which a convergent sequence approaches its limit is called the rate of convergence...

ln(x) isn't a convergent series, it's a function.

and as mentioned before, the series expansion for ln(x) only converges for a small range of x.

 

[petermer]

Ok, I certainly agree with that. But it is a fact that the function lnx is a very slow function, meaning it converges to infinity (as x goes to infinity) with a very slow rate. I understand that I do not have a series here, but would like to know if there is a similar method to the rate of convergence for functions.

 

[Irrational]

well if $$y = \ln x$$, then i guess that $$dy/dx = 1/x$$ so evaluting the limit of this would give 0... which makes sense if you interpret what $$\ln x$$ looks like graphically.

as expected, the rate of change of the function would slow down to a point where it's basically not changing as x goes to infinity.