Is there a total summation function? I define a summation function to be a partial function $F$ from infinite sequences of real numbers to the extended reals, such that:
(1) Sequences which are zero in all but possibly one position are assigned the value in that position.
(2) If $F(X) = c$ and $F(Y) = d$, and $c$ and $d$ are not opposite infinities, then $F(X+Y)= c+d$.
(3) $F(cX)$ = $cF(X)$ , for $c$ a real number.
(4) If $X$ and $Y$ are two sequences such that $Y$ is $X$ with the insertion of one term $y$, then either $F(X)$ and $F(Y)$ are both undefined or $F(X) + y = F(Y)$.
Is there a summation function which is defined on all sequences of real numbers?
 A: I don't know whether there is such a total function; however, assuming its existence, say $F=\varSigma$, we can spot some well-defined values. For example, $$\varSigma(1,a,a^2,a^3,...)=\frac{1}{1-a}\;\;\text{for}\;1\neq a\in\Bbb R,$$$$\varSigma(F_0,F_1,F_2,...)=-1,$$$$\varSigma(1,-2,3,-4,...)=\frac14,$$where $F_0=0,F_1=1,...$ are the Fibonacci numbers. Disappointingly for some, perhaps, $\varSigma(1,2,3,...)$ does not equal $-\frac{1}{12}$, but rather is $\pm\infty$ (I don't know which sign, but it looks as though the choice is arbitrary). The basic infinite result is $\varSigma(1,1,1,...)=\pm\infty$, and I guess that this extends to sequences $(p(0),p(1),p(2),...),$ where $p(\cdot)$ is polynomial.
The displayed results are clearly by analytic extension of the elementary formulae for the geometric series and its derivatives. I don't know how far we can get by pushing analytic extension in other ways; but it plainly doesn't work for $\zeta(-1)$.
I set out to look for a contradiction, but all I could find were nice consistent results. Perhaps anything that is heading for a contradiction will end up in the permitted $\{-\infty,\infty\}$ "sin bin", where it can be conveniently kept aside. 
