The improvement of the convergence properties of a Series, also called Convergence Acceleration, such
that a Series reaches its limit to within some accuracy with fewer terms than required before.
Convergence improvement can be effected by forming a linear combination with a Series whose sum is
known. Useful sums include

(1) | |||

(2) | |||

(3) | |||

(4) |

Kummer's transformation takes a convergent series

(5) |

(6) |

(7) |

(8) |

Euler's Transform takes a convergent alternating series

(9) |

(10) |

(11) |

Given a series of the form

(12) |

(13) |

(14) |

where

(15) |

(16) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 16, 1972.

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 288-289, 1985.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. *HAKMEM.* Cambridge, MA: MIT
Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Flajolet, P. and Vardi, I. ``Zeta Function Expansions of Classical Constants.'' Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps.

© 1996-9

1999-05-26