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We say that the sequence converges with order $q$ to $L$ for $q>1$ if $ \lim_{k\to \infty} \frac{|x_{k+1}-L|}{|x_k-L|^q} = \mu, \quad\; \mu > 0.$

If $a_k$ converges with order $p$, $b_k$ converges with order $q$ and $p, can we prove $b_k$ converges faster than $a_k$?

By converging faster I mean $ \lim_{k\to\infty}\frac{|b_k-L|}{|a_k-L|}=0 $

If it was the same sequence, I can prove convergence with order $q$ implies convergence with order $p$. However, here are two sequences and I don't know how to prove it.

Does higher convergence order guarantee higher convergence speed? Thank you.

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