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

$A=\sum _{n=1}^{\infty }a_{n}$

be an infinite sum whose value we wish to compute, and let

$B=\sum _{n=1}^{\infty }b_{n}$

be an infinite sum with comparable terms whose value is known. If the limit

$\gamma :=\lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}$

exists, then $a_{n}-\gamma \,b_{n}$ is always also a sequence going to zero and the series given by the difference, $\sum _{n=1}^{\infty }(a_{n}-\gamma \,b_{n})$, converges. If $\gamma \neq 0$, this new series differs from the original $\sum _{n=1}^{\infty }a_{n}$ and, under broad conditions, converges more rapidly. We may then compute $A$ as

$A=\gamma \,B+\sum _{n=1}^{\infty }(a_{n}-\gamma \,b_{n})$,

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,

$A=\gamma \,B+\sum _{n=1}^{\infty }(1-\gamma \,b_{n}/a_{n})\,a_{n}$.

Example

Consider the Leibniz formula for π:

$1\,-\,{\frac {1}{3}}\,+\,{\frac {1}{5}}\,-\,{\frac {1}{7}}\,+\,{\frac {1}{9}}\,-\,\cdots \,=\,{\frac {\pi }{4}}.$

We group terms in pairs as

$1-\left({\frac {1}{3}}-{\frac {1}{5}}\right)-\left({\frac {1}{7}}-{\frac {1}{9}}\right)+\cdots$

$\,=1-2\left({\frac {1}{15}}+{\frac {1}{63}}+\cdots \right)=1-2A$

where we identify

$A=\sum _{n=1}^{\infty }{\frac {1}{16n^{2}-1}}$.

We apply Kummer's method to accelerate $A$, which will give an accelerated sum for computing $\pi =4-8A$.

Let

$B=\sum _{n=1}^{\infty }{\frac {1}{4n^{2}-1}}={\frac {1}{3}}+{\frac {1}{15}}+\cdots$

$\,={\frac {1}{2}}-{\frac {1}{6}}+{\frac {1}{6}}-{\frac {1}{10}}+\cdots$

This is a telescoping series with sum value 1⁄2. In this case

$\gamma :=\lim _{n\to \infty }{\frac {\frac {1}{16n^{2}-1}}{\frac {1}{4n^{2}-1}}}=\lim _{n\to \infty }{\frac {4n^{2}-1}{16n^{2}-1}}={\frac {1}{4}}$

and so Kummer's transformation formula above gives

$A={\frac {1}{4}}\cdot {\frac {1}{2}}+\sum _{n=1}^{\infty }\left(1-{\frac {1}{4}}{\frac {\frac {1}{4n^{2}-1}}{\frac {1}{16n^{2}-1}}}\right){\frac {1}{16n^{2}-1}}$

$={\frac {1}{8}}-{\frac {3}{4}}\sum _{n=1}^{\infty }{\frac {1}{16n^{2}-1}}{\frac {1}{4n^{2}-1}}$

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 $(2n)^{2}$,

$\pi =4-8A$

$=3+6\cdot \sum _{n=1}^{\infty }{\frac {1}{(4(2n)^{2}-1)((2n)^{2}-1)}}$

$=3+{\frac {2}{15}}+{\frac {2}{315}}+{\frac {6}{5005}}+\cdots$

See also

• Euler transform

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. ISBN 9780486318615.
• Conrad, Keith. "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.

External links

• Weisstein, Eric W. "Kummer's Series Transformation". MathWorld.