Convergence vs. Asymptotics

This is a confusing point (and distinction).

Convergence:
- a sequence of functions $f_k$ -converges- to $f$
  iff it converges in some sense:
  - pointwise
  - uniformly
  - compact-open
  - etc.
  Note that these all imply -pointwise- convergence
  -at every point in the domain-.

Asymptotic:
A function $f_k$ is an asymptotic approximation of $f$ iff
- $f_k - f \to 0$
- or $f_k/f \to 1$ 
as $x$ goes to
- 0
  [or any other finite point, but WLOG, $0$]
- infinity
Asymptotics are all about the -germ- of a function:
the value on some -neighborhood- of $0$ or $\infty$.
In particular, a better asymptotic may be a -worse- approximation at
some other points: convergence can be very slow.

-------------------------
Taylor series are -especially- confusing b/c they mix the two of these:
+ a Taylor polynomial is an asymptotic,
  and the optimal polynomial asymptotic
  [among polynomials of a given degree]
  [and this requires $n$-times differentiability;
   indeed, this -characterizes- differentiability;
   see comments in "taylor/characterizes"]

That's the content of the theorem
$$f(x)-P_n(x) \in o(x^n)$$
(and more generally $f(x)-P_{n,a}(x-a) \in o((x-a)^{n+1})$):
the degree $n$ Taylor approximation is:
- an -asymptotic- for $f$ -at zero- (or at $a$)
  of goodness $o(x^n)$
- indeed it's the best -asymptotic- among degree $n$ polys
  [easy corollary]

+ The Taylor polynomials -converges- to a -formal power series-,
  namely the Taylor series.
  However, the Taylor series needn't have a positive radius of
  convergence and thus needn't define a function.
  Further, even if it does define a function, this function need have
  -nothing- to do with the original function.

  However, if (and only if) the original function is analytic on a
  disc (in the complex plane), the Taylor polynomials converge
  compact-open to the function on that disc.
...the error term is given in Taylor's other theorem.
  Note that the convergence can be very slow:
  just b/c you're the best asymptotic at $0$ doesn't mean you're a
  very good estimate at $1/2$.

Which matters more?
For applications, the -asymptotics- is far more important.
The connection to analytic functions is fundamental for complex
analysis, but is not important for most users.
[In particular, physicists are using -asymptotics-;
 they're not really assuming that functions are globaly analytic
 unless they're nuts or know something I don't.]

-------------------------
Some notes on asymptotics, culled from "Monthly" v111 #3,
review of "Asymptotology".

One should distinguish:
- asymptotics
- applied math
- divergence
- lack of rigor
[and - being bad/worthless]

Consider the expansion
$$x! \sim \sqrt{2\pi x} (x/e)^x\left[1+\frac{1}{12x} +\frac{1}{288x^2}
 + \dotsb\right]$$
If you -fix- $x$ and let the terms go to infinity, it -diverges for all $x$-.
Oops.
But if you fix a number of terms, and let $x$ get large, it gives a
great answer: $f(x)/x! \to 1$ rather quickly.
So in particular, the first term is Stirling's approximation, and
adding the second term makes it even better, and actually better for
$x=1$. But adding the third term makes it worse for $x=1$ (and
eventually it'll diverge).

Asymptotologists quip:
``divergent series are better than convergent ones, because they
  converge faster''
See, this approximation diverges for fixed $x$, but for fixed number
of terms, converges faster as $x$ varies.

``an approximate formula is better than an exact formula''
Exact formulae are often inscrutable; an approximate formula tells you
`what the function looks like' (in terms of familiar functions)
