   Chapter 11, Problem 53RE

Chapter
Section
Textbook Problem

# Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan−1 x, and ln(1 + x).53. f ( x ) = 1 / 16 − x 4

To determine

To find: The Maclaurin series for given function 116x4 and its radius of convergence.

Explanation

Given:

The function is 116x4 .

Result used:

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

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

The radius of convergence (1+x)k is 1.

Calculation:

The given function is expressed as,

116x4=116(1x16)4=12(1x16)14=12(1x16)14

Substitute x16 for x and 14 for k in equation (1),

116x4=12n=1(14n)(x16)n=12[1+(14)(x16)+(14)(141)2!(x16)2+(14)(141)(142)3!(x16)3                                           ++14(141)(14(n1))n!(x16)n+]

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