A formal power series can be loosely thought of as a polynomial with infinitely many terms.

In other words, a formal power series is just an object that records a sequence of coefficients.

In a sense, all formal power series are Taylor series.

An important operation on formal power series is coefficient extraction.

Thus, in these respects formal power series behave like Taylor series.

Fields of positive characteristic, they are formal power series in variable over a one-dimensional local field, i. e.

The formula is also valid for formal power series and can be generalized in various ways.

We represent this by the following formal power series in X:

He also did much work using generating functions, treated as formal power series, without concern for convergence.

In the general case these are formal power series with possibly infinite coefficients, and have to be interpreted accordingly.