# Can you check if the convergence is correct?

So I have the following examples:

1. $x_k={1 \over 4^k}; k=1,2,3,\ldots$
2. $x_k={1 \over k+1}; k=0,1,2,\ldots$
3. $x_k={1 \over (k+1)^2}; k=1,2,3,\ldots$

I found that these are all linearly convergent.

Can someone tell me if i did something wrong, or give me better convergence proof?

The answer should be one of these following: linear, R-linear, quadratic, or superlinear

Thanks

Take a look at this definitions:

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I think you'll find that the other three notions all imply linear convergence, so it could be that you are correct that your examples are all linearly convergent, while at the same time some of them might also be, say, quadratically convergent. – Gerry Myerson Nov 19 '12 at 2:12
Indeed, you might take it as an exercise to prove that R-linear convergence implies quadratic convergence, which implies superlinear convergence, which implies linear convergence. – Gerry Myerson Nov 19 '12 at 2:14

For #3, $1/(k+1)^2$ divided by $1/k^2$ doesn't go to zero, so the convergence is not superlinear; it is merely linear.
For #1, compare it to the definition of R-linear convergence, and I think you'll have no trouble finding appropriate values of $M$ and $c$.
Aren't these two sublinearly convergent since $c=1$? – Erick Wong Nov 19 '12 at 3:58
@Erick, yes. I guess I read into it that $c$ could depend on $k$, but that's not the way it's written. – Gerry Myerson Nov 19 '12 at 5:31