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 the limit exists, then
is always also a sequence going to zero and the series given by the difference,
, converges.
, this new series differs from the original
and, under broad conditions, converges more rapidly.
, the terms can be written as the product
, the sum is over a component-wise product of two sequences going to zero, 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 1⁄2.
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
and involves a fastly converging sum over just the squared even numbers
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