The inverse tangent function admits the series representation x = _ k=0 ^ (-1)^k x^…

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

Problem

The inverse tangent function admits the series representation

\arctan x = \sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{2k+1},\qquad |x|\le 1.

Answer the following questions concerning numerical approximations derived from this series. (a) Verify that, for

|x|\le 1

, the absolute values of consecutive non-zero terms of the series decrease whenever

k\ge 0

. Clearly indicate any inequalities you employ. (b) For

x=\tfrac13

, determine the least integer

n

such that truncating the series after the

n

-th non-zero term guarantees an absolute error below

10^{-5}

. Provide the resulting numerical approximation of

\arctan(\tfrac13)

rounded to eight decimal places. (c) Let

S_3(x)

denote the degree-3 polynomial obtained by keeping the first two non-zero terms of the series (through

x^{3}

). Identify all real numbers

x

[-1,1]

for which the absolute error

|\arctan x-S_3(x)|

is smaller than

0.01

. Present your solution as an interval or union of intervals, and justify every inequality used in your reasoning.