Differentiation and Integration of power series

Differentiation and Integration of power series
Since a power series is a function, it is natural to ask if the function is continuous, differentiable or integrable. The following theorem answers this question.
Term-by-Term Differentiation of Power series:
A theorem tells us that a power series can be differentiated term by term at each point in the interior of its interval of convergence.

The Term-by-Term Differentiation Theorem:
, it for some converges for If
defines a function f :
,
Such a function f has derivatives of all orders inside the interval of convergence. We can obtain the derivatives by differentiating the original series term by term:




and so on. Each of these derived series converges at every interior point of the interval of convergence of the original series.
Example:
Consider the series
, hence the center of convergence This series converges for
0 and the radius is 1. By the above theorem

has center of convergence 0 and radius of convergence 1 also.



Term-by-Term Integration of Power series:
Another theorem states that a power series can also be integrated term by term throughout its interval of convergence.

The Term-by-Term Integration Theorem:
for some converges for If
, then the series for
and converges for


for
Example:
Consider the series
, hence the center of convergence This series converges for
0 and the radius is 1. By the above theorem
has center of convergence 0 and radius of convergence 1 also.
Addition and Subtraction of Power Series:
Suppose that we have two functions and their power series representations
and
then


Multiplication of Power Series:
Still another theorem states that power series can be multiplied term by term.


The series Multiplication Theorem for Power Series:
, converges absolutely for and If both
and

, also converges absolutely for Then the series
and