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

Then, is differentiable by theorem 27.1.7, and


(ii)
 Consider the power series
 


The power series is absolutely convergent (by ratio test) for

It is divergent for and convergent for . Thus

is defined. Since the serie

is convergent for and divergent for , we have
.
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