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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: enter image description here

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    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.2012-11-19

1 Answers 1

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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 $c$.

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    @Erick, yes. I guess I read into it that $c$ could depend on $k$, but that's not the way it's written.2012-11-19