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

摘要

Sums of the form ∑ν=1xf(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=1-1/21ν=-2ln2.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 limn→∞(2n)-n2/2-n/4e-n/8∏ν=12nν(-1)νν2/4=e7ζ(3)/16π2.