Given a sequence, there are -discrete- versions of calculus one can do
on it:
- "discrete differentiation"
  taking successive differences
- "discrete integration"
  summing as terms in a series

The analogies go a rather long way; indeed, the study of this and
generalizations to other time domains goes by such names as "discrete
calculus" and "time scales", and is an active area of research:
EG Martin Bohner, "Dynamic Equations on Time Scales"
In 1988, someone figured out how to take derivatives on general time
scales on their thesis, and around 2000 or so they developed a chain
rule. A fundemantal theorem is easy, and hence we have integration as
anti-derivatives, but a good theory of -integration- (like Riemann
integration) is hard.

Apparently the results are elegant, but the proof techniques are
rather ugly so far.

More generally, every time you see an analogy, there should be an
overarching theory.

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(what if you have an "interpolating function", i.e.,
 a continuous/smooth/etc. function $f(x)$ such that for integers, $f(n)=a_n$.
(hmmm...better yet, an analytic function)).
EG:
- discrete differentiation is slopes of secant lines
  (I use this in the example of $2^x$ frequently)
- the integral test for convergence of monotone series
  (especially for $\zeta(p)$ for $p\geq 1$, aka ``$p$-series'')

Note that the -analytic- properties of an interpolating function
reflect properties of the series they interpolate.
EG growth rate of functions: derivatives of continuous functions
  reflect the growth of a sequence they interpolate
EG integral test for series:
  integrals of functions

In particular, Bernoulli numbers (see Spivak's calculus, chapter 27,
for an interesting exercise) arose from examining the operator
$Af(x)= f(x+1)-f(x)$,
which is exactly the ``discrete differentiation'' operator
  (technically ``forward differentiation'', since it's $+1$)
(and I'd guess that scaled versions of this operator converge
 to the derivative operator)


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quick intro in American Math Monthly, Dec 2001:
"How to Differentiate and Integrate Sequences", Donald R. Chalice

Further notes:
(forward) Difference equations are the discrete analog to
differential equations; a difference equation being one involving the
operator $\Delta f(n) = f(n+1)-f(n)$.

discrete derivatives/integrals: analogies with continuous ones:
 - fundamental theorem (obvious here)
   $\sum_{i=0}^n \Delta A(i) = \left. A(i)\right|_0^{n+1} = A(n+1) - A(0)$
   ...note the $n+1$ (and note how Knuth et.~al.\ deal with it in
   ``Concrete Mathematics'', by changing the notational meaning of $\sum_0^n$)
 - two sequences with discrete deriv. equal clearly just differ by
   constant
 - increasing iff deriv positive, etc.
 - interesting special cases:
  - polynomials
    (derivatives easy binomial theorem; integral is summation
  formulae!)

indeed, here the correct basis of polynomials is:
$p_k(n)=n(n-1)\dots(n-k+1)$
by $\delta p_k = kp_{k-1}$
  note that $p_k(n)$ is $n$ permute $k$, and you can see the difference
  formula combinatorially:
  $p_k(n+1)-p_k(n)=kp_{k-1}(n)$
  an ordered selection of $k$ things from $n+1$
  either includes the last one (at any of $k$ points), leaving $k-1$
    choices of the remaining $n$ (hence $kp_{k-1}(n)$)
  or doesn't, leaving $k$ choices from $n$, hence $p_k(n)$.
you also get taylor series and taylor polynomials:
$A(n+l) \approx A(n)+\Delta^1 A(n) p_1(l) + \dots + \frac{\Delta^k A(n)}{k!}p_k(l)$
(note that it holds on the nose for $l\leq k$, I think...)
  (and you get an error term if you'd like: it's what you think it
   should be)
...and polynomials are characterized by $\Delta^k f(n)\equiv 0$.

Note that in this way, one can derive all the sums of $k$-th powers
rules: express $x^k$ in terms of the elementary polynomials, and
integrate!

  - $2^x$ (and other exponents): derivative is itself
  - bump (0 0 0 0 1 0 0 0 0) ($1$ concentrated at $0$) -- derivatives "look" like derivative of
    bump, (also known as ``impulse'')
    and are given by binomial formula with sign
...so many results in continuous setting are kinda "limits" of these
obvious ones

further, $\delta^k A = (-1)^k A * \delta^k \delta$, where $\delta$ is
  the bump. Note that this is exactly the same as in the continuous
  case, where $f^{(k)} = f * \delta^{(k)}$.
hence $\delta^1 \delta = (1 -1 0 0 0)$
and $\delta^2 \delta = (1 -2 1 0 0)$ etc.
whence $\delta^2 A(n) = A(n+2) - 2A(n+1) + A(n)$ etc.
----------------------
In general time scales, the classical Leibnitz rule isn't quite true:
$(fg)'(x)=f'(x)g(x)+f(x)g'(x)$
but for instance
$\Delta (fg)(n)=\Delta f(n) g(n)+f(n+1)\Delta g(n)
    =\Delta f(n) g(n+1)+f(n)\Delta g(n)
    \neq \Delta f(n) g(n)+f(n)\Delta g(n)$
The trouble is that the little square in the corner of the graphical
proof of the product rule doesn't disappear.

In general, you need to consider the graininess operator $\sigma$
  (i.e., how big the next jump is),
and writing $f^\sigma(x) :=f(x+\sigma(x))$, we obtain:
$(fg)^\Delta=f^\Delta g + f^\sigma g^\Delta
=f^\Delta g^\sigma + f g^\Delta$

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There's also apparently two notions of ``standard polynomial'', and
hence you get two version of Taylor's theorems.
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Linear recursive series (eg, Fibonnaci) of length $k$ arise as order
$k$ linear difference equations (in strong analogy to order $k$ linear
differential equations).
