   Chapter 11.10, Problem 85E

Chapter
Section
Textbook Problem

Use the following steps to prove (17).(a) Let g ( x ) = ∑ n = 0 ∞ ( k n ) x n . Differentiate this series to show that g ′ ( x ) = k g ( x ) 1 + x     − 1 < x < 1 (b) Let h ( x ) = ( 1 + x ) − k g ( x ) and show that h ′ ( x ) = 0 .(c) Deduce that g ( x ) = ( 1 + x ) k .

To determine

(a)

To show:

g'x=kgx1+x

Explanation

1) Concept:

Binomial series:

If  k  is any real number and x<1, then

1+xk=n=0knxn

2) Given:

gx=n=0knxn

3) Calculation:

It is given that

gx=n=0knxn

Differentiate with respect to x

g'x=n=1knnxn-1

Multiply by 1+x to both sides

1+xg'x=1+xn=1knnxn-1

=n=1knnxn-1+n=0knnxn

Replace n  by n+1 in the first series

=n=0kn+1n+1xn+

To determine

(b)

To show:

h'x=0

To determine

(c)

To deduce:

gx=1+xk

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