   Chapter 11.10, Problem 24E

Chapter
Section
Textbook Problem

Find the Taylor series for f ( x ) centered at the given value of a. [Assume that f has a power series expansion. Do not show that R n ( x ) → 0 .] Also find the associated radius of convergence. f ( x ) = cos x , a = π / 2

To determine

To find:

Taylor series for   f(x), and find its associated radius of convergence

Explanation

1) Concept:

Taylor series of the function   f at   a is

fx=n=0fnan!(x-a)n=fa+f'a1!x-a+f''a2!x-a2+f'''a3!x-a3+

The ratio test:

(i) If limnan+1an=L<1, then the series n=1an is absolutely convergent.

(ii) If limnan+1an=L>1 or limnan+1an=, then the series n=1an is divergent.

(iii) If limnan+1an=1, the ratio test is inconclusive, that is, no conclusion can be draw about the convergence or divergence of an.

2) Given:

fx=cosx

3) Calculation:

As   a=π2, Taylor series is given by:

fx=n=0fnπ2n!x-π2n=fπ2+f'π21!x-π2+f''π22!x-π22+f'''π23!x-π23+

Let’s find the coefficients of this series.

fx=cosx

So,

fπ2=cosπ2=0

Differentiate   f(x) with respect to x.

f'x=-sinx

So,

fπ2=-sinπ2=-1

Now, differentiate   f'(x) with respect to x to get f''x.

f''x=-(cosx)

f''x=-cosx

So,

f''π2=-cosπ2=0

Now, differentiate f''x with respect to x to get f'''x.

f'''x=--sinx

f'''x=sinx

So,

f'''π2=sinπ2=1

Now, differentiate f'''x with respect to x to get f''''x.

f''''x=cosx

So,

f''''π2=cosπ2=0

Now, differentiate f''''x with respect to x to get f(5)x.

f(5)x=-sinx

So,

f(5)π2=-sinπ2=-1

Now, differentiate f(5)x with respect to x to get f(6)x

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