power series

For small x, the power series expansion is given above.

Moreover there can be no other power series with this property.

Figure 4 shows the relative errors using the power series.

It treats the case where the power series are formal.

Let be a function which corresponds to a power series.

A limitation of the power series solution shows itself in this example.

Then can be expanded into a power series and inverted.

In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves.

From this, one can extrapolate to obtain other power series.

In other cases the quantile functions may be developed as power series.

In particular it may be defined by the following power series:

This means that g can be written in a power series of the following form:

The transition between the two materials can usually be approximated by means of a power series.

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

An important operation on formal power series is coefficient extraction.

One approach to construct regular functions is to use power series with real coefficients.

The above result can be derived by power series expansion of .

It is particularly useful in connection with power series.

This equation may be solved by several methods, including the classical power series approach.

Formal power series are often used merely to represent the whole collection of their coefficients.

The power series defined below is generalized by the idea of a germ.

This means that each function is expanded as a power series in .

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

The most common type of asymptotic expansion is a power series in either positive or negative terms.

The power series may converge slowly or even not converging when the energy levels are close to each other.