Series acceleration

As convergence acceleration is defined as the replacement of a sequence by another which converges faster to the same limit. These methods are often used to calculate values ​​of rows.

One consequence of

With the limit converges faster than another episode

With the same threshold value, if the limit

Exists and is zero. Obtained from a convergent sequence by a sequence transformation of the form

We speak of convergence acceleration.

Example

The sequence converges with an error proportional to against. The terms in the sum can be used for k> 1 by

Be estimated. The series for the various estimates the left and right are telescopic series,

The difference is the last two terms

This also applies

The n - th row of the partial sums occurring therein converging with errors proportional to so much faster.

This process can be continued as desired, so the difference of the last row to the telescope range may be considered.