Let and denote for each integer .

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

Problem

Let

p_n(x)=\dfrac{d^{n}}{dx^{n}}\bigl(\arctan x\bigr)

and denote

a_n=p_n(0)

for each integer

n\ge 0

. (a) Compute

a_0,\,a_1,\,a_2,\,a_3,

and

a_4

exactly. (b) Conjecture a closed‐form expression for

a_{2k}

and for

a_{2k+1}

, where

k

is any non-negative integer. State each conjecture clearly. (c) Prove the conjectured formulas from part (b) using mathematical induction on

k

. (d) Using your result, write the first four non-zero terms of the Maclaurin series for

\arctan x

and specify the general term. (e) Determine the radius of convergence of the Maclaurin series found in part (d).