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Accelerated convergence is the term used to describe the replacement of a sequence by another that converges more quickly to the same limit value .

There are a number of different methods of accelerating convergence to choose from depending on the properties of the original sequence. Typical applications are iterative calculations, the evaluation of series and integration ( Romberg method ).

definition

One episode

${\ displaystyle T = \ {t_ {n} \} _ {n \ in \ mathbb {N} _ {0}}}$

with the limit converges faster than any other sequence
${\ displaystyle s}$

${\ displaystyle S = \ {s_ {n} \} _ {n \ in \ mathbb {N} _ {0}}}$

exists and is zero. Obtained from a convergent sequence by a sequence transformation of the shape
${\ displaystyle T}$${\ displaystyle S}$

${\ displaystyle T = F (S)}$,

so one speaks of convergence acceleration.

example

The sequence converges with the order of convergence as against . The asymptotic development applies${\ displaystyle a_ {n} = \ sum _ {k = 1} ^ {n} {\ frac {1} {k ^ {2}}}}$${\ displaystyle {\ frac {1} {n}}}$${\ displaystyle {\ frac {\ pi ^ {2}} {6}}}$

The n -th partial sum of the series occurring in it converges with the order of convergence as , so much faster.
${\ displaystyle {\ frac {1} {n ^ {3}}}}$

This process can be continued at will, so the difference between the last row and the telescope row can be observed.
${\ displaystyle \ sum _ {k = 2} ^ {\ infty} {\ frac {1} {(k-1) k (k + 1) (k + 2)}}}$