Consider the series

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

Problem

Consider the series

\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}

. Scientist X will compute the partial sum

S_N

until the alternating-series error bound falls below

0.05

. Scientist Y will instead stop at

N-1

terms and report the midpoint

M_N = \dfrac{S_{N-1}+S_N}{2}

. Find the minimum

N

required for each scientist to achieve the stated accuracy and specify whose strategy requires fewer term evaluations.