Editing Rate of convergence (section)

===Examples===

The [[geometric progression]] <math display="inline">(a_k) = 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \ldots, 1/{2^k}, \dots </math>  converges to <math>L = 0</math>. Plugging the sequence into the definition of Q-linear convergence (i.e., order of convergence 1) shows that

<math display="block">\lim_{k \to \infty} \frac{\left| 1/2^{k+1} - 0\right|}{\left| 1/ 2^k - 0 \right|} = \lim_{k \to \infty} \frac{2^k}{2^{k+1}} = \frac{1}{2}. </math>

Thus <math>(a_k)</math> converges Q-linearly with a convergence rate of <math>\mu = 1/2</math>; see the first plot of the figure below.

More generally, for any initial value <math>a</math> in the real numbers and a real number common ratio <math>r</math> between -1 and 1, a geometric progression <math>(a r^k)</math> converges linearly with rate <math>|r|</math> and the sequence of partial sums of a [[geometric series]] <math display="inline">(\sum_{n=0}^k ar^n)</math> also converges linearly with rate <math>|r|</math>. The same holds also for geometric progressions and geometric series parameterized by any [[Complex number|complex numbers]] <math>a \in \mathbb{C}, r \in \mathbb{C}, |r| < 1.</math>

The staggered geometric progression <math display="inline">(b_k) = 1, 1, \frac{1}{4}, \frac{1}{4}, \frac{1}{16}, \frac{1}{16}, \ldots, 1/4^{\left\lfloor \frac{k}{2} \right\rfloor}, \ldots,</math> using the [[Floor_and_ceiling_functions|floor function]] <math display="inline">\lfloor x \rfloor</math> that gives the largest integer that is less than or equal to <math>x,</math> converges R-linearly to 0 with rate 1/2, but it does not converge Q-linearly; see the second plot of the figure below. The defining Q-linear convergence limits do not exist for this sequence because one subsequence of error quotients starting from odd steps converges to 1 and another subsequence of quotients starting from even steps converges to 1/4. When two subsequences of a sequence converge to different limits, the sequence does not itself converge to a limit. Generally, for any staggered geometric progression <math>(a r^{\lfloor k / m \rfloor})</math>, the sequence will not converge Q-linearly but will converge R-linearly with rate <math display="inline">\sqrt[m]{|r|}; </math> these examples demonstrate why the "R" in R-linear convergence is short for "root."

The sequence 
<math display="block">(c_k) = \frac{1}{2}, \frac{1}{4}, \frac{1}{16}, \frac{1}{256}, \frac{1}{65,\!536}, \ldots, \frac{1}{2^{2^k}}, \ldots</math>
converges to zero Q-superlinearly. In fact, it is quadratically convergent with a quadratic convergence rate of 1. It is shown in the third plot of the figure below.

Finally, the sequence 
<math display="block">(d_k) = 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \ldots, \frac{1}{k + 1}, \ldots</math>
converges to zero Q-sublinearly and logarithmically and its convergence is shown as the fourth plot of the figure below.

[[Image:ConvergencePlots.png|thumb|alt=Plot showing the different rates of convergence for the sequences ''a''<sub>''k''</sub>, ''b''<sub>''k''</sub>, ''c''<sub>''k''</sub> and ''d''<sub>''k''</sub>.|Log-linear plots of the example sequences ''a''<sub>''k''</sub>, ''b''<sub>''k''</sub>, ''c''<sub>''k''</sub>, and ''d''<sub>''k''</sub> that exemplify linear, linear, superlinear (quadratic), and sublinear rates of convergence, respectively.|600px|center]]