Module 9 :  Infinite Series, Tests of Convergence,  Absolute and Conditional Convergence, Taylor and Maclaurin Series
Lecture 27 :  Series of functions [Section 27.1]
 

For a power series

if is the interval of convergence, then for every , let

.

Then

is a function on the interval . The properties of this function are given by in the next theorems, which we assume without proof.

27.1.7 Theorem (Differentiation of power series) :
 

Let a power series

have non-zero radius of convergence and

Then, the following holds:

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