Define the series

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

Problem

Define the series

\displaystyle \sum_{n=1}^{\infty} (-1)^{n+1}\,\dfrac{(\tfrac13)^{n}}{n}

. Let

P

denote its exact sum. (a) Determine the least positive integer

N

for which the absolute error

|P-S_{N}|

is guaranteed to be less than

10^{-5}

, where

S_{N}

is the

N

th partial sum. (b) Using the

N

you found in part (a), compute

S_{N}

to six decimal places. (c) Establish a numerical interval of length less than

2\times10^{-5}

that must contain

P

. Write your answer in interval notation.