   Chapter 11.10, Problem 32E

Chapter
Section
Textbook Problem

# Use the binomial series to expand the function as a power series. State the radius of convergence.32. 8 + x 3

To determine

To expand: The power series of given function: State the radius of convergence.

Explanation

Given:

The function is 8+x3 .

Result used:

(1) The Binomial series: If k is any number and |x|<1 then, (1+x)k can be written as

(1+x)k=n=1(kn)xn=1+kx+k(k1)2!x2+k(k1)(k2)3!+ (1)

(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:

The given function is expressed as follows,

8+x3=8(1+x8)3=2(1+x8)13

Substitute x8 for x and 13 for k in equation (1),

(8+x)3=2n=1(13n)(x8)n=2[1+13(x8)+(13)(131)2!(x8)2+13(131)(132)3!(x8)3  +13(131)(132)(133)4!(x8)4++13(131)(13(n1))n!(x8)n+]                                =2[1+124x+13(133)2!(x8)2+13(133)(163)3!(x8)3 +13(133)(163)(193)4!(x8)4++13(133)(1(3n3)3)n!(x8)n+]=2[1124x+(1)2322!(x8)2+(1)225333!(x8)3+(1)3258344!(x8)4+

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