   Chapter 11.10, Problem 15E

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 Rn(x) → 0.] Also find the associated radius of convergence.15. f(x) = 2x

To determine

To find: The Maclaurin series for f(x) by using the definition of a Maclaurin series and also the radius of the convergence.

Explanation

Given:

The function is f(x)=2x .

Result used:

(1) "The expansion of the Maclaurin series f(x)=n=0f(n)(0)n! is f(0)+f01!x+f(0)2!x2+f(0)3!x3+

(2) The Ratio Test:

“(i) If limn|an+1an|=L<1 , then the series n=1an is absolutely convergent (and therefore convergent.)

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

(ii) If limn|an+1an|=1 , the Ratio Test inconclusive; that is, no conclusion can be drawn about the convergence or divergence of n=1an .

Calculation:

Obtain f(0) .

Substitute 0 for x in f(x) .

f(0)=20=1

Find the first derivative of f(x) at a=0 .

f(x)=ddx(2x)=ddx(eln2x)=ddx(exln2)=exln2ddx(xln2)

That is, f(x)=2x(ln2) (1)

Substitute 0 for x,

f(0)=20(ln2)=(1)(ln2)=ln2

Find the second derivative of f(x) at a=0 .

f(2)(x)=d2dx2(f(x))=ddx(f(x))  =ddx(2x(ln2))    (by equation(1))=(ln2)(ln2)2x    [ddx(2x)=2x(ln2)]

Simplify further and obtain f(2)(0) as shown below.

f(2)(x)=2x(ln2)2 (2)

f(2)(0)=20(ln2)2=(1)(ln2)2=(ln2)2

Find the third derivative of f(x) at a=0 .

f(3)(x)=d3dx3(f(x))=ddx(f(2)(x))=ddx(2x(ln2)2)    (by equation(2))=(ln2)2(ln2)2x

Simplify further and obtain f(3)(0) as shown below.

f(3)(x)=(ln2)32x (3)

f(3)(0)=(ln2)320=(ln2)3(1)=(ln2)3

Find the fourth derivative of f(x) at a=0

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