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 12. In this case

and Kummer's transformation gives

This simplifies to

which converges much faster than the original series.

See also

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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