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Kummer's transformation of series

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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.

Technique

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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 the limit

exists, then is always also a sequence going to zero and the series given by the difference, , converges. If , this new series differs from the original and, under broad conditions, converges more rapidly.[1] We may then compute as

,

where is a constant. Where , the terms can be written as the product . If for all , the sum is over a component-wise product of two sequences going to zero,

.

Example

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Consider the Leibniz formula for π:

We group terms in pairs as

where we identify

.

We apply Kummer's method to accelerate , which will give an accelerated sum for computing .

Let

This is a telescoping series with sum value 12. In this case

and so Kummer's transformation formula above gives

which converges much faster than the original series.

Coming back to Leibniz formula, we obtain a representation of that separates and involves a fastly converging sum over just the squared even numbers ,

See also

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References

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  1. ^ Holy et al., On Faster Convergent Infinite Series, Mathematica Slovaca, January 2008
  • 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. ISBN 9780486318615.
  • Conrad, Keith. "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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