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 asExample
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.
In this case
and Kummer's transformation gives
This simplifies to
which converges much faster than the original series.
