   Chapter 11.10, Problem 34E

Chapter
Section
Textbook Problem

# Use the binomial series to expand the function as a power series. State the radius of convergence.34. (1 − x)3/4

To determine

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

Explanation

Given:

The function is (1x)34 .

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:

Substitute x for x and 34 for k  in equation (1),

[1+(x)]34=n=1(34n)(x)n=(1+34(x)+(34)(341)2!(x)2+34(341)(342)3!(x)3+34(341)(342)(343)4!(x)4+34(341)(34(n1))n!(x)n+)=(134x+(344)2!x234(344)(384)3!x3+34(344)(384)(3124)4!x4++34(344)(34(n1)4)n!xn+)=(134x+(14)2!x234(14)(54)3!x3+34(14)(54)(94)4!x4++3(344)(34(n1)4)4n!xn+)

Simplify further and obtain the series,

[1+(x)]34=134

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