   Chapter 11.10, Problem 13E

Chapter
Section
Textbook Problem

Find the Maclaurin series for f ( x ) using the definition of a Maclaurin series. [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

To determine

To find:

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

Explanation

1) Concept:

Maclaurin series of the function   f,

fx=n=0fn0n!(x)n=f0+f'01!x+f''02!x2+f'''03!x3+

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:

Maclaurin series is given by:

fx=n=0fn0n!(x)n=f0+f'01!x+f''02!x2+f'''03!x3+

Let’s find the coefficients of this series

fx=cosx

So,

f0=cos0=1

Differentiate   f(x) with respect to x.

f'x=-sinx

So,

f0=-sin0=0

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

f''x=-(cosx)

f''x=-cosx

So,

f''0=-cos0=-1

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

f'''x=--sinx

f'''x=sinx

So,

f'''0=sin0=0

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

f''''x=cosx

So,

f''''0=cos0=1

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

f(5)x=-sinx

So,

f(5)0=-sin0=0

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

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