Let and let be the th partial sum.

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

Problem

Let

S=\displaystyle\sum_{n=1}^{\infty} (-1)^{n+1}\dfrac1n

and let

S_N=\displaystyle\sum_{n=1}^{N} (-1)^{n+1}\dfrac1n

be the

N

th partial sum. (a) Verify that the sequence

a_n=\dfrac1n

is positive, decreasing, and has limit

0

n\to\infty

. (b) Determine the least positive integer

N

for which the approximation

S\approx S_N

is guaranteed to have an error of magnitude less than

0.001

. Justify your answer. (c) Using the value of

N

you found in part (b), compute

S_N

to four decimal places. (d) Show explicitly that the remainder

|S-S_N|

satisfies the inequality required in part (b).