Kummer’s acceleration method

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

n=1an=S. (1)

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

We suppose that the terms an of (1) are nonzero.  Let


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

limnanbn=ϱ 0

and the identity

S=ϱC+n=1(1-ϱbnan)an. (2)

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

limn(1-ϱbnan)= 0.

Example.  For replacing the series


by a faster converging series we may take


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

C=n=1(1n-1n+1)= 1.

Now we have  ϱ=1,  and using (2) we obtain

S= 1+n=11n2(n+1).

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


where now  C=14;  then we get


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


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


  • 1 Pascal Sebah & Xavier Gourdon: http://numbers.computation.free.fr/Constants/constants.htmlAcceleration of the convergence of series (2002).
Title Kummer’s acceleration method
Canonical name KummersAccelerationMethod
Date of creation 2014-12-12 10:34:19
Last modified on 2014-12-12 10:34:19
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type AlgorithmMathworldPlanetmath
Classification msc 26A06
Classification msc 40A05
Related topic ValueOfTheRiemannZetaFunctionAtS2