Ways to sum

We gave -a- definition of the ``sum'' of a sequence
(as the limit of the sequence of partial sums).

However, there's other ways of doing it!

To axiomatize, a ``summation'' should be:
- a functional on sequences
  (like, duh)
- sorry, make that ``on some subspace of sequences'',
  called ``the summable sequences''
- it should agree with regular ``summation''
  on compactly supported sequences
  [IE, finitely many non-zero terms]

- frankly, it should also agree on -absolutely summable-
  sequences [IE, $L^1$ -- the totally unambiguous stuff]

We might ask for it to be a ``good extention'' in some way;
as it stands, it allows all sorts of junk.
Perhaps you want it to respect some sorta topology.
[I think if you use the following topology, you get
 partial sums:
 - a sequence $(a_i)_n$ (of sequences!) converges to $a$
   iff for all $N$, there's some point $M_N$ after which
   $(a_i)_n = a_i$ for $i<N, n > M_N$.
   Urk -- correct analysis definition, unreadable.
   I -don't- mean: if you look at each coordinate,
     eventually it's constant.
   I mean ``eventually the initial segment stabilizes''.
   More canonically, ``eventually every subset stabilizes''
   ...and then you approximate by finite sums?
I'm a little skeptical...]

Note that this is a huge jump in abstraction:
we're talking about partial functionals on infinite-dim'l vector
spaces!

It's also unnecessary right now;
it's worth noting that we made a -choice-,
and some of the reasons it's an ``okay choice'',
but delving into more detail is not very useful for much math,
though it is naturally interesting.

That is, it's an illustration of trying to -axiomatize-;
of trying to get our hands around a -concept-.

Cesaro sums/abel summation
(like $1-1+1-1+\dotsb = 1/2$)
is useful in studying analytic functions (so the real analysis
pathologies you get in complex analysis)
but that's about it for now.

