For the Maclaurin series of , (1+x)=x- x^ 2 2 + x^ 3 3 - x^ 4 4 + , -1<x 1

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

Problem

For the Maclaurin series of

\ln(1+x)

\ln(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots,\qquad -1<x\le 1,

consider approximating

\ln(1+0.5)

using a truncated series. Throughout this problem an "approximation of degree

n

" refers to the polynomial consisting of the first

n

non-zero terms of the Maclaurin series above. (a) Write explicitly the polynomial of degree

4

that approximates

\ln(1+x)

for

x=0.5

. Evaluate this polynomial numerically to four decimal places. (b) Show that, for

0<x\le 1

, the magnitudes of the successive non-zero terms of this series form a decreasing sequence that tends to

0

. Provide a clear argument based on algebraic comparison. (c) Determine the smallest positive integer

n

such that the absolute error in approximating

\ln(1+0.5)

by the degree-

n

polynomial is less than

10^{-4}

. State both the chosen value of

n

and the numerical approximation obtained to six decimal places.