Improved formulas for complete and partial summation of certain series.

Abstract

Abstract. In two previous articles one of the authors gave formulas, with numerous examples, for summing a series either to'infinity (complete) or up to a certain number n of terms (partial) by considering the sum of the first j terms Sj, or some suitable modification Sj, closely related to Sj, as a polynomial in 1/j. Either «S«, or Sn was found by m-point Lagrangian extrapolation from S¡t , S¡t-i > • • • , Sjt-m+i to 1/j = 0 or 1/j = 1/n respectively. This present paper furnishes more accurate m-point formulas for sums (or sequences) S¡ which behave as even functions of 1/j. Tables of Lagrangian extrapolation coefficients in the variable 1/j are given for: complete summation, m = 2(1)7, ja = 10, exactly and 20D, m = 11, jo = 20, 30D; partial summation, m = 7,> = 10, n = 11(1)25(5)100, 200, 500, 1000, exactly. Applications are made to calculating ir or the semi-perimeters of many-sided regular polygons, Euler's constant,

Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

“…A more complete discussion of the properties of the linear but nonregular sequence transformation Λ (n) k (β, s n ) can be found in articles by Salzer [66,67], Salzer and Kimbro [68], and Wimp [69] as well as in Wimp's book (see pp. 35 -38 of ref.…”