Improved formulas for complete and partial summation of certain series.

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 ${S_j}$, or some suitable modification ${S’_j}$, closely related to ${S_j}$, as a polynomial in 1/j. Either ${S_\infty }$ or ${S_n}$ was found by m-point Lagrangian extrapolation from ${{S}_{{{j}_{0}}}}$, ${{S}_{{{j}_{0}}-1}}$, $\cdot \cdot \cdot$, ${{S}_{{{j}_{0}}-m+1}}$ to 1/j = 0 or 1/j = 1/n respectively. This present paper furnishes more accurate m-point formulas for sums (or sequences) ${S_j}$ which behave as even functions of 1/j. Tables of Lagrangian extrapolation coefficients in the variable $1/{j^2}$ are given for: complete summation, m = 2(1)7, ${j_0}$ = 10, exactly and 20D, m = 11, ${j_0}$ = 20, 30D; partial summation, m = 7, ${j_0}$ = 10, n = 11(1)25(5)100, 200, 500, 1000, exactly. Applications are made to calculating $\pi$ or the semi-perimeters of many-sided regular polygons, Euler’s constant, \[ 1 + \sum \limits _{r = 1}^j {\left \{ {\tfrac {{ - 1}}{{{{(4r - 1)}^2}}} + \tfrac {1}{{{{(4r + 1)}^2}}}} \right \} = 1 - \tfrac {1}{{{3^2}}} + \tfrac {1}{{{5^2}}} - \cdots } {\text {for}} j = \infty \] (Catalan’s constant), calculation of a definite integral as the limit of a suitably chosen sequence, determining later zeros of ${J_v}(x)$ from earlier zeros for suitable v, etc. A useful device in many cases involving sums of odd functions, is to replace ${S_j}$ by a trapezoidal-type ${S’_j}$ which is seen, from the Euler-Maclaurin formula, to be formally a series in $1/{j^2}$. In almost every example, comparison with the earlier (1/j)-extrapolation showed these present formulas, for 7 points, to improve results by anywhere from around 4 to 9 places.