How to add a non-integer number of terms, and how to produce unusual infinite summations

Abstract

Sums of the form ∑ ν = 1 x f ( ν ) are defined traditionally only when the number of terms x is a positive integer or ∞ . We propose a natural way to extend this definition to the case when x is a (rather arbitrary) real or complex number (“fractional sums”). This generalizes known special cases like the interpolation of the factorial by the Γ function, or Euler's little-known formula ∑ ν = 1 - 1 / 2 1 ν = - 2 ln 2 . After giving the fundamental definition, we generalize several algebraic identities (such as the geometric series) to the case with a non-integer number of terms. We use these ideas to derive a number of unusual infinite sums, products and limits, such as lim n → ∞ ( 2 n ) - n 2 / 2 - n / 4 e - n / 8 ∏ ν = 1 2 n ν ( - 1 ) ν ν 2 / 4 = e 7 ζ ( 3 ) / 16 π 2 .