  We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x
f(\nu) \] when the number of terms x is a rather arbitrary real or even complex
number. The resulting theory includes the known interpolation of the factorial
by the gamma function or Euler's little-known formula \sum_{\nu=1}^{-1/2} \frac
1\nu = -2\ln 2.
  Many classical identities like the geometric series and the binomial theorem
nicely extend to this more general setting. Sums with a fractional number of
terms are closely related to special functions, in particular the Riemann and
Hurwitz zeta functions. A number of results about fractional sums can be
interpreted as classical infinite sums or products or as limits, including
identities like \[ \lim_{n\to\infty}[ e^{\frac n 4(4n+1)}n^{-\frac 1 8 -
n(n+1)}(2\pi)^{-\frac n 2} \prod_{k=1}^{2n} \Gamma(1+\frac k 2)^{k(-1)^k} ] =
\sqrt[12]{2} \exp({5/24} - \frac 3 2 \zeta'(-1) -\frac{7\zeta(3)}{16\pi^2}) \]
some of which seem to be new.
