z-adic Summation

Formal power series

Let

f = c₀ + c₁ z + c₂ z² + c₃ z³ + ...

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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