Consider the series expansion for

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the series expansion

\arctan x = x-\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}-\dfrac{x^{4}}{4}+\cdots

for

|x|\le 1

. A student wishes to use the alternating nature of the series to bound the error of a polynomial approximation at

x=0.5

. Which of the following conditions most directly justifies that the magnitudes of the terms decrease, allowing the bound to be applied? (a)

\dfrac{x^{k+1}}{k+1}<\dfrac{x^{k}}{k}

for all

k\ge 1

when

|x|<1

. (b)

\dfrac{|x|^{k+1}}{k+1}<\dfrac{|x|^{k}}{k}

for all

k\ge 1

when

|x|\le 1

. (c)

|x|^{k+1}<|x|^{k}

for all

k\ge 1

when

|x|<1

. (d)

\dfrac{|x|^{k+1}}{k+1}>\dfrac{|x|^{k}}{k}

for all

k\ge 1

when

|x|<1