Editing Rate of convergence (section)

=== Examples ===
For any two [[Geometric progression|geometric progressions]] <math>(a r^k)_{k \in \mathbb{N}}</math> and <math>(b s^k)_{k \in \mathbb{N}},</math> with shared limit zero, the two sequences are asymptotically equivalent if and only if both <math>a = b</math> and <math>r = s.</math> They converge with the same order if and only if <math>r = s.</math> <math>(a r^k)</math> converges with a faster order than <math>(b s^k)</math> if and only if <math>r < s.</math> The convergence of any [[geometric series]] to its limit has error terms that are equal to a geometric progression, so similar relationships hold among geometric series as well. Any sequence that is asymptotically equivalent to a convergent geometric sequence may be either be said to "converge geometrically" or "converge exponentially" with respect to the absolute difference from its limit, or it may be said to "converge linearly" relative to a logarithm of the absolute difference such as the "number of decimals of precision." The latter is standard in numerical analysis.

For any two sequences of elements proportional to an inverse power of <math>k,</math> <math>(a k^{-n})_{k \in \mathbb{N}}</math> and <math>(b k^{-m})_{k \in \mathbb{N}},</math> with shared limit zero, the two sequences are asymptotically equivalent if and only if both <math>a = b</math> and <math>n = m.</math> They converge with the same order if and only if <math>n = m.</math> <math>(a k^{-n})</math> converges with a faster order than <math>(b k^{-m})</math> if and only if <math>n > m.</math> 

For any sequence <math>(a_k)_{k \in \mathbb{N}}</math> with a limit of zero, its convergence can be compared to the convergence of the shifted sequence <math>(a_{k-1})_{k \in \mathbb{N}},</math> rescalings of the shifted sequence by a constant <math>\mu,</math> <math>(\mu a_{k-1})_{k \in \mathbb{N}},</math> and scaled <math>q</math>-powers of the shifted sequence, <math>(\mu a_{k-1}^q)_{k \in \mathbb{N}}.</math> These comparisons are the basis for the Q-convergence classifications for iterative numerical methods as described above: when a sequence of iterate errors from a numerical method <math>(|x_k - L|)_{k \in \mathbb{N}}</math> is asymptotically equivalent to the shifted, exponentiated, and rescaled sequence of iterate errors <math>(\mu |x_{k-1} - L|^q)_{k \in \mathbb{N}},</math> it is said to converge with order <math>q</math> and rate <math>\mu.</math>