Kummer's transformation of series
In mathematics, specifically in the field of numerical analysis, Kummer's transformation of series is a method used to accelerate the convergence of an infinite series. The method was first suggested by Ernst Kummer in 1837.
Let
be an infinite sum whose value we wish to compute, and let
be an infinite sum with comparable terms whose value is known. If
then A is more easily computed as
Example
We apply the method to accelerate the Leibniz formula for π:
First group terms in pairs as
where
Let
which is a telescoping series with sum 1⁄2. In this case
and Kummer's transformation gives
This simplifies to
which converges much faster than the original series.
gollark: Continuation passing style quicksort in a hilariously slow interpreter.
gollark: It manages *1* second, which is great.
gollark: When writing osmarkslisp™, I cared about performance to the extent that it would sort a list of 200 integers in under 5 seconds.
gollark: `b"%s"`?
gollark: Maybe a similar thing to LTSes, where you need to scan through it to find the point where some constraints are satisfied, but checking that requires solving linear equations or something.
See also
- Euler transform
References
- Senatov, V.V. (2001) [1994], "Kummer transformation", Encyclopedia of Mathematics, EMS Press
- Knopp, Konrad (2013). Theory and Application of Infinite Series. Courier Corporation. p. 247.
- Keith Conrad. "Accelerating Convergence of Series" (PDF).
- Kummer, E. (1837). "Eine neue Methode, die numerischen Summen langsam convergirender Reihen zu berech-nen". J. Reine Angew. Math. (16): 206–214.
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