Kummer's transformation of series

Let

$$A\; =\; \backslash sum\_\{n=1\}^\backslash infty\; a\_n$$

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

$$B\; =\; \backslash sum\_\{n=1\}^\backslash infty\; b\_n$$

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

If the limit

$$\backslash gamma\; :=\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash frac\{a\_n\}\{b\_n\}$$

exists, then $a\_n-\backslash gamma\; \backslash ,b\_n$ is always also a sequence going to zero and the series given by the difference, $\backslash sum\_\{n=1\}^\backslash infty\; (a\_n\; -\; \backslash gamma\backslash ,\; b\_n)$, converges.

If $\backslash gamma\; \backslash neq\; 0$, this new series differs from the original $\backslash sum\_\{n=1\}^\backslash infty\; a\_n$ and, under broad conditions, converges more rapidly.Holy et al., [https://www.researchgate.net/publication/26512388_On_Faster_Convergent_Infinite_Series On Faster Convergent Infinite Series], Mathematica Slovaca, January 2008

We may then compute $A$ as

$$A\; =\; \backslash gamma\backslash ,B\; +\; \backslash sum\_\{n=1\}^\backslash infty\; (a\_n\; -\; \backslash gamma\backslash ,b\_n),$$

where $\backslash gamma\; B$ is a constant. Where $a\_n\backslash neq\; 0$, the terms can be written as the product $(1-\backslash gamma\backslash ,b\_n/a\_n)\backslash ,a\_n$.

If $a\_n\backslash neq\; 0$ for all $n$, the sum is over a component-wise product of two sequences going to zero,

$$A=\backslash gamma\backslash ,B\; +\; \backslash sum\_\{n=1\}^\backslash infty\; (1-\backslash gamma\backslash ,b\_n/a\_n)\backslash ,a\_n$$.

Consider the Leibniz formula for π:

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

We group terms in pairs as

$$\backslash begin\{align\}$$

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

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

\end{align}

where we identify

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

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

Let

$$\backslash begin\{align\}$$

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

\end{align}

This is a telescoping series with sum value {{frac|1|2}}.

In this case

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

and so Kummer's transformation formula above gives

$$\backslash begin\{align\}$$

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}

\end{align}

which converges much faster than the original series.

Coming back to Leibniz formula, we obtain a representation of $\backslash pi$ that separates $3$ and involves a fastly converging sum over just the squared even numbers $(2n)^2$,

$$\backslash begin\{align\}$$

\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

\end{align}

• Euler transform

{{Reflist}}

{{refbegin}}

• {{SpringerEOM|title=Kummer transformation|id=Kummer_transformation|last=Senatov|first=V.V.}}
• {{cite book

| last = Knopp

| first = Konrad

| title = Theory and Application of Infinite Series

| url = https://books.google.com/books?id=ac_DAgAAQBAJ&pg=PA247

| year = 2013

| publisher = Courier Corporation

| page = 247

| isbn = 9780486318615

}}

• {{cite web | url = https://kconrad.math.uconn.edu/blurbs/analysis/series_acceleration.pdf | title = Accelerating Convergence of Series | first = Keith | last = Conrad }}
• {{cite journal

|last1= Kummer

|first1=E.

|year= 1837

|title=Eine neue Methode, die numerischen Summen langsam convergirender Reihen zu berech-nen

|journal= J. Reine Angew. Math.

|issue= 16

|pages= 206–214

|url=https://archive.org/details/journalfrdierei23crelgoog/page/n215

}}

{{refend}}

• {{MathWorld|title = Kummer's Series Transformation|urlname = KummersSeriesTransformation}}

Category:Series acceleration methods

{{mathanalysis-stub}}