From the way linear/quadratic/cubic convergence of a sequence are defined, I wonder why they are called linear/quadratic/cubic, in the sense of some connections to linear/quadratic/cubic functions.
Here are the definitions of linear/quadratic/cubic convergence of a sequence in my words based on Wikipedia
Suppose that the sequence $\{x_k\}$ converges to the number $L$. Suppose $q > 1$.
When $\lim_{k\to \infty} \frac{|x_{k+1}-L|}{|x_k-L|^q} = μ$ and $μ ∈ (0, 1)$, we say that the sequence (Q-)converges linearly if $q=1$, quadratically if $q=2$, and cubically if $q=3$.
Similarly, how is logarithmic convergence connected to a logarithm function? The definition of logarithmic convergence is from the same link to Wikipedia:
If the sequences converges sublinearly and additionally $ \lim_{k\to \infty} \frac{|x_{k+2} - x_{k+1}|}{|x_{k+1} - x_k|} = 1, $ then it is said the sequence $\{x_k\}$ converges logarithmically to $L$.
I found a plot of linear, linear, quadratic and logarithmic rates of convergence for an example in Wikipedia, which seems to suggest some connection, although it is not clear to me how they are connected:
Thanks for clarification!