Let and consider its Maclaurin series expansion .

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

Problem

Let

f(x)=\arctan x

and consider its Maclaurin series expansion

f(x)=\sum_{n=0}^{\infty}(-1)^{n}\dfrac{x^{2n+1}}{2n+1}

. (a) Prove that the sequence of term magnitudes

a_n(x)=\dfrac{|x|^{2n+1}}{2n+1}

is decreasing for

x

in the interval

0<|x|\le1

. (b) For

x=0.9

, determine the minimum integer

N

such that the absolute error in approximating

\arctan(0.9)

by the partial sum

P_N(0.9)=\sum_{n=0}^{N}(-1)^{n}\dfrac{0.9^{2n+1}}{2n+1}

is less than

10^{-4}

. Show all reasoning leading to your choice of

N

. (c) Using the value of

N

found in part (b), compute

P_N(0.9)

to five decimal places and state the corresponding error bound.