Consider the function

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

Problem

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

, which is analytic on

(-1,1)

. The series is centered at

a=0

(Maclaurin), and you will approximate

g(0.5)

with a Taylor polynomial. (a) For

x

[0,0.5]

and any integer

k\ge 1

, compute an explicit upper bound

M

for

|g^{(k)}(x)|

. Express

M

in a form that depends on

k

, and justify why this bound is valid on the stated interval. (b) Derive an inequality that bounds the error

|g(0.5)-T_{n}(0.5)|

, where

T_{n}

is the

n

th–degree Maclaurin polynomial. (c) Using the inequality from part (b), find the smallest integer

n

that guarantees

|g(0.5)-T_{n}(0.5)|<10^{-6}

. Fully explain each step in solving the inequality for

n

and verify that your chosen

n

satisfies the required tolerance.