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
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.
[Holy et al.]
''On Faster Convergent Infinite Series''
Mathematica Slovaca, January 2008
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
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 .
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
*
Euler transform
References
*
*
*
*
External links
*
Numerical analysis
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