The Representation of Power Series

Power series are represented in the Wolfram System as (RefLink[SeriesData, paclet:ref/SeriesData]) objects.

The power series is printed out as a sum of terms, ending with (RefLink[O, paclet:ref/O])[x] raised to a power:

Series[Cos[x],{x,0,4}]

Internally, however, the series is stored as a (RefLink[SeriesData, paclet:ref/SeriesData]) object:

InputForm[%]

By using (RefLink[SeriesData, paclet:ref/SeriesData]) objects, rather than ordinary expressions, to represent power series, the Wolfram System can keep track of the order and expansion point, and do operations on the power series appropriately. You should not normally need to know the internal structure of (RefLink[SeriesData, paclet:ref/SeriesData]) objects.

You can recognize a power series that is printed out in standard output form by the presence of an (RefLink[O, paclet:ref/O])[x] term. This term mimics the standard mathematical notation O(x), and represents omitted terms of order x. For various reasons of consistency, the Wolfram System uses the notation (RefLink[O, paclet:ref/O])[x]^n for omitted terms of order (x^n), corresponding to the mathematical notation ((O(x))^n), rather than the slightly more familiar, though equivalent, form O((x^n)).

Any time that an object like (RefLink[O, paclet:ref/O])[x] appears in a sum of terms, the Wolfram System will in fact convert the whole sum into a power series.

The presence of (RefLink[O, paclet:ref/O])[x] makes the Wolfram System convert the whole sum to a power series:

a x+Exp[x]+O[x]^3

Series objects can involve fractional powers:

Series[Sqrt[1-x^3],{x,1,5}]

Here is the internal representation of the series:

%//InputForm

Series can involve logarithmic terms:

Series[x^x,{x,0,4}]

The logarithmic factors appear explicitly inside the (RefLink[SeriesData, paclet:ref/SeriesData]) coefficient list:

%//InputForm