The function has the Maclaurin series for

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

The function

g(x)=\ln(1+x)

has the Maclaurin series

\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\dfrac{x^{n}}{n}

for

-1<x\le 1

. You wish to approximate

g(0.6)

using a Taylor polynomial of degree

N

. (a) Express the remainder

R_{N}(0.6)=g(0.6)-P_{N}(0.6)

in terms of an integral involving

g^{(N+1)}(c)

for some

c

between

0

and

0.6

. No evaluation is required. (b) Obtain a numerical bound on

|R_{N}(0.6)|

that comes from the alternating character of the series. Your bound should be written explicitly in terms of

N

. (c) Derive a second numerical bound on

|R_{N}(0.6)|

that relies on a maximum value of

|g^{(N+1)}(x)|

[0,0.6]

. Express the bound in closed form in terms of

N

. (d) Determine the smallest integer

N

for which both bounds ensure that

|R_{N}(0.6)|<1\times10^{-4}

. State which of the two bounds gives the more efficient guarantee and explain your reasoning.