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## Homework Statement

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

"

*Suppose {β[itex]_{n}[/itex]}from n=1 → ∞ is a sequence known to converge to zero, and [itex]\alpha[/itex][itex]_{n}[/itex] converges to a number [itex]\alpha[/itex]. If a positive constant K exists with*

|[itex]\alpha_{n} - \alpha|[/itex]≤K|β[itex]_{n}[/itex]|, for large n, then we say that [itex]\alpha[/itex][itex]_{n}[/itex] converges to [itex]\alpha[/itex] with a rate of convergence O(β[itex]_{n}[/itex])

|[itex]\alpha_{n} - \alpha|[/itex]≤K|β[itex]_{n}[/itex]|, for large n, then we say that [itex]\alpha[/itex][itex]_{n}[/itex] converges to [itex]\alpha[/itex] with a rate of convergence O(β[itex]_{n}[/itex])

## Homework Equations

β[itex]_{n}[/itex]=1/n[itex]^{p}[/itex]

## 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.