   Chapter 11.R, Problem 52E

Chapter
Section
Textbook Problem

Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for e x , sin x , tan − 1 x , and ln ( 1 + x ) . f ( x ) = 10 x

To determine

To find:

i) The Maclaurin series

ii) The radius of convergence R of fx

Explanation

1) Concept:

i) For a power series   n=0cnx-an, there is a positive number R such that the series converges if x-a<R and diverges if  x-a>R, this number R is called the radius of convergence and the series is centred at  x=a.

ii) The Maclaurin series expansion for  ex.

ex=n=0xnn!, for all x

iii) The ratio test:

If limnan+1an=L<1, then the series an is absolutely convergent, and

if limnan+1an=L>1, then the series diverges.

2) Given:

fx=10x

3) Calculation:

It is given that,

fx=10x

It can be written as, using the rules of logarithm,

10x=eln10x

Replace x  by  ln10x   in the Maclaurin series expansion for  ex.

eln10x=n=0ln10xnn!

=n=0ln10nxnn!

10x=n=0ln10nxnn!

This is the Maclaurin series expansion for  10x

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