Other notions of convergence, for divergent series:
(see: Hardy, "Divergent Series", for lotta info on this)

Suppose $\sum a_n$ diverges -- can we still make some
reasonable sum?

- Cesaro sum
  (lim of averages)
- Abel sum
  If $\sum a_n z^n$ has radius of convergence $1$,
  then we can look at $\lim_{z\to 1^-} a_n z^n$.
NB: general philosophy: embed a question into a context with more structure
  and math
  (replace a series by a power series; a sequence by a generating function,
   etc.)
...finding converses to this (when does a sum exist, if an Abel sum exists?)
leads to Tauberian theorems, and modern analysis.

(Euler, Hoelder, Borel, Gauss, Poisson, Weierstrass had some other ideas...)
(e.g., Euler's way of doing it works on the geometric series for all $\Re(z)<1$)
- Banach limit?
  (look in exercises following Hahn-Banach thm.)


In both these senses, $1-1+1-1+\dotsb=\frac{1}{2}$, for instance.

I guess the right way to say this is:
"We're giving various extensions of the summation functional
 from the subspace of convergent series to bigger subspaces of
 $\Hom(\bN,\bC)$"
...presumably these are continuous extensions?

another way would be "axiomatically", i.e., qua power series etc.
just require:
- additivity
- homogeneity
  (i.e., taken together = linearity)
- shift independence:
  $\sum_{n=0}^\infty a_n = a_0 + \sum_{n=1}^\infty a_n$
...this works for any power series that defines an analytic function,
 I think... (monodromy questions??)
 e.g., geometric series

