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?