Operations on Power Series

For the power series

the operations listed in the table below will produce another power series.

Thus from a given power series we can produce many more power series
using these operations.

(The examples and exercises in the text use these five operations on power series, often one after another.)
 Name of Operation

 

 Description of the Operation

(The operations in bold are the most useful ones.)
   
1. Add Add (or subtract)  like terms to a  given power series
(1) a constant
(2) a term of the form c(x-a)n
(3) a polynomial in (x-a)
(4) a power series expanded about a.
2. Multiply Multiply (or sometimes divide by) like terms 
(1) a constant
(2) a term of the form c(x-a)n          (sometimes applicable for division)
(3) a multinomial in (x-a)                                 (not applicable for division)
(4) a power series expanded about a.        (not applicable for division)
3. Replace Replace (x-a) by c(mx+b)p and simplify to a power series expanded about another value.
4. Differentiate Differentiate term by term with respect to x
5. Integrate Integrate term by term with respect to x to obtain either
(1) an indefinite integral with an additive constant or
(2) a definite integral