# Kummer’s acceleration method

There are several methods for acceleration of the convergence of a given series

 $\displaystyle\sum_{n=1}^{\infty}a_{n}\;=\;S.$ (1)

One of the simplest is the following one due to Kummer (1837).

We suppose that the terms $a_{n}$ of (1) are nonzero.  Let

 $\sum_{n=1}^{\infty}b_{n}\;=\;C$

be a series with nonzero terms and the known sum $C$.  We use the limit

 $\lim_{n\to\infty}\frac{a_{n}}{b_{n}}\;=\;\varrho\;\neq\;0$

and the identity

 $\displaystyle S\;=\;\varrho C+\sum_{n=1}^{\infty}\left(1-\varrho\frac{b_{n}}{a% _{n}}\right)a_{n}.$ (2)

Thus the original series (1) has attained a new form (2) the convergence of which is faster because of

 $\lim_{n\to\infty}\left(1-\varrho\frac{b_{n}}{a_{n}}\right)\;=\;0.$

Example.  For replacing the series

 $\sum_{n=1}^{\infty}\frac{1}{n^{2}}\;=\;S$

by a faster converging series we may take

 $\sum_{n=1}^{\infty}\frac{1}{n(n\!+\!1)}\;=:\;C,$

which, for its part, can be expressed as the telescoping series

 $C\;=\;\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n\!+\!1}\right)\;=\;1.$

Now we have  $\varrho=1$,  and using (2) we obtain

 $S\;=\;1+\sum_{n=1}^{\infty}\frac{1}{n^{2}(n\!+\!1)}.$

The convergence of this series may accelerated similarly taking e.g.

 $\sum_{n=1}^{\infty}\frac{1}{n(n\!+\!1)(n\!+\!2)}\;=:\;C,$

where now  $C=\frac{1}{4}$;  then we get

 $S\;=\;\frac{5}{4}+2\!\sum_{n=1}^{\infty}\frac{1}{n^{2}(n\!+\!1)(n\!+\!2)}.$

The procedure may be repeated $N$ times in all, yielding the result

 $S\;=\;\sum_{n=1}^{N}\frac{1}{n^{2}}+N!\sum_{n=1}^{\infty}\frac{1}{n^{2}(n\!+\!% 1)(n\!+\!2)\cdots(n\!+\!N)}.$

As for the sum of this series, see Riemann zeta function at $s=2$ (http://planetmath.org/valueoftheriemannzetafunctionats2).

## References

• 1 Pascal Sebah & Xavier Gourdon: http://numbers.computation.free.fr/Constants/constants.htmlAcceleration of the convergence of series (2002).
Title Kummer’s acceleration method KummersAccelerationMethod 2014-12-12 10:34:19 2014-12-12 10:34:19 pahio (2872) pahio (2872) 8 pahio (2872) Algorithm msc 26A06 msc 40A05 ValueOfTheRiemannZetaFunctionAtS2