One can do other things with sequence than just ask if they converge!
(it's like students w/quadratics: they often just -immediately- apply
the quadratic formula. Yes, this is the first thing to do, but not the
-only- thing to do!!)

For instance, if a sequence goes off to infinity, you can ask about
its rate of growth. Conversely, if it converges, you can ask how
quickly it converges.
(these are both basically growth/big-O stuff)

Spaces of sequences -----------------
(orient yourself, just like spaces of functions like $C^2$ let you
orient yourself)
Other than sequences going off to infinity, all our sequences will
live inside $l^\infty$.
$l^\infty > CS > l^1$
$l^\infty > l^2 > l^1$
(and of course $l^\infty > CV > l^p$ for all $p<\infty$)
(EG to show this is tight: $a_n=n^{-1/p}$)
($CS=$ conditionally summable/convergent: a kinda weird space)
($CV=$ convergent sequences; contained within Caesaro convergent)
$l^1=$ absolutely summable, i.e., $L^1(\bN)$

Can also do Caesaro (sp?) convergent:
$\lim \frac{1}{n}(a_1+\dots+a_n)$
basically, the natural domain for this limit of functionals:
a particular extention of ``lim'' to a bigger space

Incidentally, the closure of "eventually constant" functions in
$l^\infty$ is "convergent sequences" -- cute, eh?

Maps between spaces of sequences ---------
derivative/difference
integral/partial sums
(so is polynomial growth saying that some derivative is convergent?)
(can also do Taylor series/polynomials)

Other things to do --------------
image of sequence
(and other stuff: sup of image, inf of image)

limit of image (woah!!! your first inverse limit!)
  (i.e., $\set{a_n,a_{n+1},\dotsc}$)
lim sup, lim inf
(note that these are not ``sup lim'' or ``inf lim'': you take sup/inf
 of each image, and take the limit of these, which exist b/c they're
 monotone)

functionals --------------
note that lim sup and lim inf are not linear...they're perhaps
"sublinear", i.e., $\limsup a_n + b_n \leq \limsup a_n + \limsup b_n$
(like an absolute value), and behaves well w/r/t multiplication by
-positive- numbers -- but not negative! that wrecks havoc!

there's lots of functionals on these spaces
there's also lots of operators, e.g.,
$l^\infty \times l^1 \to l^1$
(and more generally $l^p \times l^q \to l^1$??)
(apparently also multiplication by $1/n$ takes $l^2$ into $l^1$?)

natural question: do all functionals factor as an operator followed by
one specific functional?
(i.e., operators are a semigroup that act (on the right) on functionals by
precomposition; what are the "orbits" under this action?)

