Kummer’s acceleration method
There are several methods for acceleration of the convergence of a given series
One of the simplest is the following one due to Kummer (1837).
We suppose that the terms of (1) are nonzero. Let
be a series with nonzero terms and the known sum . We use the limit
and the identity
Thus the original series (1) has attained a new form (2) the convergence of which is faster because of
Example. For replacing the series
by a faster converging series we may take
which, for its part, can be expressed as the telescoping series
Now we have , and using (2) we obtain
The convergence of this series may accelerated similarly taking e.g.
where now ; then we get
The procedure may be repeated times in all, yielding the result
As for the sum of this series, see Riemann zeta function at (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|
|Date of creation||2014-12-12 10:34:19|
|Last modified on||2014-12-12 10:34:19|
|Last modified by||pahio (2872)|