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Computing Finite Sums

In: Summability Calculus

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  • Ibrahim M. Alabdulmohsin

    (King Abdullah University of Science and Technology)

Abstract

In this chapter, we derive methods for computing the values of fractional finite sums. The motivation behind the methods developed in this chapter is twofold. First, the Euler-Maclaurin summation formula and all of its analogs diverge too rapidly and, hence, they cannot be used to calculate finite sums, not even with the help of summability methods. Second, computing the Taylor series expansions and using them, in turn, to calculate fractional finite sums by rigidity is a tedious task and is far from being a satisfactory approach. In this chapter, we resolve those limitations by providing a simple direct method of computing fractional finite sums. We show how well-known historical results, such as Euler’s infinite product formula for the gamma function, fall as particular cases of these more general results. We also show why the Euler-Maclaurin summation formula is connected to polynomial approximation. Finally, we extend results to composite finite sums.

Suggested Citation

  • Ibrahim M. Alabdulmohsin, 2018. "Computing Finite Sums," Springer Books, in: Summability Calculus, chapter 0, pages 115-131, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-74648-7_6
    DOI: 10.1007/978-3-319-74648-7_6
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