Let be an alternating series with and for every

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let

\displaystyle \sum_{k=1}^{\infty} (-1)^{k+1} a_k

be an alternating series with

a_k>0

and

a_{k+1}<a_k

for every

k

. If

S_N

and

S_{N+1}

are the

N

th and

(N+1)

st partial sums, respectively, which of the following statements about their accuracy as approximations of the actual series sum

S

is always true? (a)

|S-S_N|<|S-S_{N+1}|

because

a_{N+1}

is positive. (b)

|S-S_{N+1}|<|S-S_N|

because the error bound for

S_{N+1}

a_{N+2}

, which is smaller than

a_{N+1}

. (c)

|S-S_N|=|S-S_{N+1}|

because consecutive partial sums straddle the exact value. (d) The relative accuracy of

S_N

and

S_{N+1}

cannot be determined without computing the exact sum

S

.‎