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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, (Catalan's constant), calculation of a definite integral as the limit of a suitably chosen sequence, determining later zeros of J,(x) from earlier zeros for suitable v, etc. A useful device in many cases involving sums of odd functions, is to replace Sj by a trapezoidal-type Sj which is seen, from the Euler-Maclaurin formula, to be formally a series in 1/j'. In almost every example, comparison with the earlier (l//)-extrapolation showed these present formulas, for 7 points, to improve results by anywhere from around.4 to 9 places.

Improved formulas for complete and partial summation of certain series.