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
- [math]\displaystyle{ A=\sum_{n=1}^\infty a_n }[/math]
be an infinite sum whose value we wish to compute, and let
- [math]\displaystyle{ B=\sum_{n=1}^\infty b_n }[/math]
be an infinite sum with comparable terms whose value is known. If the limit
- [math]\displaystyle{ \gamma:=\lim_{n\to \infty} \frac{a_n}{b_n} }[/math]
exists, then [math]\displaystyle{ a_n-\gamma \,b_n }[/math] is always also a sequence going to zero and the series given by the difference, [math]\displaystyle{ \sum_{n=1}^\infty (a_n-\gamma\, b_n) }[/math], converges. If [math]\displaystyle{ \gamma\neq 0 }[/math], this new series differs from the original [math]\displaystyle{ \sum_{n=1}^\infty a_n }[/math] and, under broad conditions, converges more rapidly.[1] We may then compute [math]\displaystyle{ A }[/math] as
- [math]\displaystyle{ A=\gamma\,B + \sum_{n=1}^\infty (a_n-\gamma\,b_n) }[/math],
where [math]\displaystyle{ \gamma B }[/math] is a constant. Where [math]\displaystyle{ a_n\neq 0 }[/math], the terms can be written as the product [math]\displaystyle{ (1-\gamma\,b_n/a_n)\,a_n }[/math]. If [math]\displaystyle{ a_n\neq 0 }[/math] for all [math]\displaystyle{ n }[/math], the sum is over a component-wise product of two sequences going to zero,
- [math]\displaystyle{ A=\gamma\,B + \sum_{n=1}^\infty (1-\gamma\,b_n/a_n)\,a_n }[/math].
Example
Consider the Leibniz formula for π:
- [math]\displaystyle{ 1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \,=\, \frac{\pi}{4}. }[/math]
We group terms in pairs as
- [math]\displaystyle{ 1 - \left(\frac{1}{3} - \frac{1}{5}\right) - \left(\frac{1}{7} - \frac{1}{9}\right) + \cdots }[/math]
- [math]\displaystyle{ \, = 1 - 2\left(\frac{1}{15} + \frac{1}{63} + \cdots \right) = 1-2A }[/math]
where we identify
- [math]\displaystyle{ A = \sum_{n=1}^\infty \frac{1}{16n^2-1} }[/math].
We apply Kummer's method to accelerate [math]\displaystyle{ A }[/math], which will give an accelerated sum for computing [math]\displaystyle{ \pi=4-8A }[/math].
Let
- [math]\displaystyle{ B = \sum_{n=1}^\infty \frac{1}{4n^2-1} = \frac{1}{3} + \frac{1}{15} + \cdots }[/math]
- [math]\displaystyle{ \, = \frac{1}{2} - \frac{1}{6} + \frac{1}{6} - \frac{1}{10} + \cdots }[/math]
This is a telescoping series with sum value 1⁄2. In this case
- [math]\displaystyle{ \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} }[/math]
and so Kummer's transformation formula above gives
- [math]\displaystyle{ 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} }[/math]
- [math]\displaystyle{ = \frac{1}{8} - \frac{3}{4} \sum_{n=1}^\infty \frac{1}{16n^2-1}\frac{1}{4n^2-1} }[/math]
which converges much faster than the original series.
Coming back to Leibniz formula, we obtain a representation of [math]\displaystyle{ \pi }[/math] that separates [math]\displaystyle{ 3 }[/math] and involves a fastly converging sum over just the squared even numbers [math]\displaystyle{ (2n)^2 }[/math],
- [math]\displaystyle{ \pi=4-8A }[/math]
- [math]\displaystyle{ =3+6\cdot\sum_{n=1}^\infty \frac{1}{(4(2n)^2-1)((2n)^2-1)} }[/math]
- [math]\displaystyle{ =3 + \frac{2}{15} + \frac{2}{315} + \frac{6}{5005} + \cdots }[/math]
See also
- Euler transform
References
- ↑ Holy et al., On Faster Convergent Infinite Series, Mathematica Slovaca, January 2008
- Hazewinkel, Michiel, ed. (2001), "Kummer transformation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Kummer_transformation
- Knopp, Konrad (2013). Theory and Application of Infinite Series. Courier Corporation. p. 247. ISBN 9780486318615. https://books.google.com/books?id=ac_DAgAAQBAJ&pg=PA247.
- Conrad, Keith. "Accelerating Convergence of Series". https://kconrad.math.uconn.edu/blurbs/analysis/series_acceleration.pdf.
- Kummer, E. (1837). "Eine neue Methode, die numerischen Summen langsam convergirender Reihen zu berech-nen". J. Reine Angew. Math. (16): 206–214. https://archive.org/details/journalfrdierei23crelgoog/page/n215.
External links
- Weisstein, Eric W.. "Kummer's Series Transformation". http://mathworld.wolfram.com/KummersSeriesTransformation.html.
Original source: https://en.wikipedia.org/wiki/Kummer's transformation of series.
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