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.

Technique

Let
:$A=\sum_^\infty a_n$
be an infinite sum whose value we wish to compute, and let
:$B=\sum_^\infty b_n$
be an infinite sum with comparable terms whose value is known.
If the limit
:$\gamma:=\lim_ \frac$

exists, then $a_n-\gamma \,b_n$ is always also a sequence going to zero and the series given by the difference, $\sum_^\infty (a_n-\gamma\, b_n)$, converges.
If $\gamma\neq 0$, this new series differs from the original $\sum_^\infty a_n$ and, under broad conditions, converges more rapidly.Holy et al.
''On Faster Convergent Infinite Series''
Mathematica Slovaca, January 2008
We may then compute $A$ as

:$A=\gamma\,B + \sum_^\infty (a_n-\gamma\,b_n)$,

where $\gamma B$ is a constant. Where $a_n\neq 0$, the terms can be written as the product $(1-\gamma\,b_n/a_n)\,a_n$.
If $a_n\neq 0$ for all $n$, the sum is over a component-wise product of two sequences going to zero,

:$A=\gamma\,B + \sum_^\infty (1-\gamma\,b_n/a_n)\,a_n$.

Example

Consider the Leibniz formula for π:
:$1 \,-\, \frac \,+\, \frac \,-\, \frac \,+\, \frac \,-\, \cdots \,=\, \frac.$
We group terms in pairs as
:$1 - \left(\frac - \frac\right) - \left(\frac - \frac\right) + \cdots$
:$\, = 1 - 2\left(\frac + \frac + \cdots \right) = 1-2A$
where we identify
:$A = \sum_^\infty \frac$.
We apply Kummer's method to accelerate $A$, which will give an accelerated sum for computing $\pi=4-8A$.

Let
:$B = \sum_^\infty \frac = \frac + \frac + \cdots$
:$\, = \frac - \frac + \frac - \frac + \cdots$
This is a telescoping series with sum value .
In this case
:$\gamma := \lim_ \frac = \lim_ \frac = \frac$
and so Kummer's transformation formula above gives
:$A=\frac \cdot \frac + \sum_^\infty \left ( 1-\frac \frac \right ) \frac$
:$= \frac - \frac \sum_^\infty \frac\frac$
which converges much faster than the original series.

Coming back to Leibniz formula, we obtain a representation of $\pi$ that separates $3$ and involves a fastly converging sum over just the squared even numbers $(2n)^2$,
:$\pi=4-8A$
:$=3+6\cdot\sum_^\infty \frac$
:$=3 + \frac + \frac + \frac + \cdots$

See also

* Euler transform

References

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External links

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Numerical analysis

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