Let S= _ n=1 ^ (-1)^ n+1 1 n^ 2 +3 be an alternating series.

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

Problem

Let

S=\sum_{n=1}^{\infty}(-1)^{n+1}\frac1{n^{2}+3}

be an alternating series. (a) Prove that the sequence

\displaystyle a_n=\frac1{n^{2}+3}

is positive, decreases monotonically for

n\ge1

, and has limit

0

n\to\infty

. Clearly indicate any calculus you use. (b) Determine the smallest integer

N

such that the partial sum

S_N

approximates

S

with an error whose magnitude is less than

0.0005

. Show all work leading to the inequality that

N

must satisfy and state the value you obtain for

N

. (c) Using your value of

N

from part (b), give an upper bound for

|S-S_N|

and interpret its meaning in the context of numerical approximation.