For , the series converges to . Two plans are proposed to approximate this value to…

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

For

x=\tfrac12

, the series

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

converges to

\arctan\!\tfrac12

. Two plans are proposed to approximate this value to within

10^{-4}

: Plan A: use the partial sum

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

. Plan B: use the average

A_N = \dfrac{S_N + S_{N+1}}{2}

. Determine the least integer

N

that guarantees the required accuracy for each plan, and indicate which plan achieves the target with fewer term evaluations.