# z-adic Summation

## Formal power series

Let
f = c_0 + c_1 z + c_2 z^2
+ c_3 z^3 + ...

be a power series.

We have learned that this series has a radius of convergence that is
either a postive real number or 0 or "infinity". For example, the
geometric series (given by c_j = 1 for all j) has radius of
convergence 1, and the exponential series (given by c_j = 1/j!
for all j) has infinite radius of convergence.

The power series f determines a function f(z) for those values of z
with |z| smaller than the radius of convergence.

But a power series **may have radius of convergence 0**.

Sometimes one wants to deal with a power series formally. When that is
the case, the knowledge of a power series is nothing more nor less than
the knowledge of its sequence of "coefficients" {c_j}.

A formal power series is **not** a function.

## The formal distance between formal power series

With f the formal power series having coefficients c_j as above,
one defines
ord_z(f) = the least j with c_j not zero

and

||f|| = 2^{-e} when e = ord_z(f) .

and

d(f, g) = || f - g || .

With respect to the formal distance two things are true:

- Every formal power series is the formal limit of its sequence
of partial sums.
- Every formal power series is the formal limit of polynomials.

There is a useful analogy between formal distance (for formal power
series) and 2-adic distance
(for 2-adic integers).
It is important to note that formal summation
has no relation to ordinary summation except within the radius of
convergence of a power series.

The point here is that there is a way in which every power series
may be regarded as convergent.

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