TECHNICAL NOTES AND SHORT PAPERS Skip-Term Summation of Sequences

The literature on the summation of sequences and the finding of the sums of the corresponding series is voluminous and well known. For rapidly convergent sequences term-by-term evaluations furnish no great obstacle. Likewise, if the sequence can be matched with a sequence whose sum is known and the term-byterm difference of the two sequences converges rapidly, this difference-sequence can be evaluated by terms. For series that converge slowly, various devices have been suggested. If the integral and derivatives of the general term, expressed as a function of the term index, can be evaluated, the Euler [3]-Maclaurin [7] formula or the extended-range form of these formulas [9] can be used to find the sum of the series, or of the residue after a selected term. If a series converges sufficiently smoothly, the formulas of Lubbock [6], as modified by de Morgan [2], can be used to evaluate a partial sum of the terms of the series from those terms which are separated by a selected gap. Lubbock's formulas were revived by Sprague [12], have been discussed in Whittaker and Robinson [14], and were extended by Steffensen [13]. Lubbock's formulas all involve differences, and tables have been published by Davis [1] and others, to facilitate the evaluations. The limit of the sum of a slowly convergent infinite series has been approximated by Salzer [10], who selected the reciprocal of the sequence index as the independent variable in Lagrangian interpolation and found the approximate limit as this variable approaches zero. He selects the last m members of the sequence of partial sums from the n members which have been calculated. Simple factors were published for n = 5, 10, 15, and 20, and for m = 4, 7, and 11, where m < n. Later, Salzer [11] published formulas and coefficients for the sum of a finite series, and Horgan [4] prepared decimal tables of coefficients for finite sums. These formulas involve no differences and are convenient for many purposes. However, they involve extrapolation. The present formulas involve only interpolation. Only a few coefficients are included, as examples of the method. If a sequence of terms {nk} can be approximated by a polynomial of degree n in the index k, the values of (n + 1) arbitrarily selected terms of the sequence can be used to determine the value of every other term. If {y A is a subset of {rik}, containing (n + 1) elements, Lagrangian interpolation determines