For the series

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

Problem

For the series

P=\displaystyle\sum_{n=1}^{\infty}(-1)^{\,n-1}\dfrac{n}{3^{n}}

, let

P_k

denote the partial sum of the first

k

terms. Which of

P_{2}

P_{3}

, and

P_{4}

are guaranteed to approximate

P

with absolute error less than

10^{-4}

? List all that satisfy the requirement and justify your selection with appropriate error bounds.