Order of Convergence & Numerical Analysis

[tehdiddulator]

Homework Statement

In my book, for a class on numerical analysis, we are given the definition:

"Suppose {β$_{n}$}from n=1 → ∞ is a sequence known to converge to zero, and $\alpha$$_{n}$ converges to a number $\alpha$. If a positive constant K exists with
|$\alpha_{n} - \alpha|$≤K|β$_{n}$|, for large n, then we say that $\alpha$$_{n}$ converges to $\alpha$ with a rate of convergence O(β$_{n}$)

Homework Equations

β$_{n}$=1/n$^{p}$

The Attempt at a Solution

I'm just looking for how I would go about trying to compute the rate of convergence. I understand that you are suppose to compare the series (One problem I am working on is sin(1/n), and the answer is sin(1/n) converges to zero as fast as (1/n) converges to zero), but I do not know how to show the algebra to get there.

 

[Jufro]

Well, let's look at the definition:

Lim_{k$\rightarrow$$\infty$} |a_k -$\alpha$|/|a_k -$\alpha$|^q < μ

Here q is the rate of convergence and μ $\in$ (0,1).

Can you use your information now trying to see what is appropriate?