Consider the convergent series with sum

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

Problem

Consider the convergent series

\displaystyle\sum_{k=1}^{\infty} a_k

with sum

S

. Which of the following best explains why the difference

|S-S_n|

can be made arbitrarily small by choosing a sufficiently large value of

n

? (a) Because the series is alternating and the magnitude of the first neglected term bounds the error. (b) Because the individual terms

a_k

approach

0

k\to\infty

. (c) Because the sequence of partial sums

\{S_n\}

approaches a finite limit as

n\to\infty

. (d) Because the successive ratios

\dfrac{a_{k+1}}{a_k}

are eventually less than

1

, so the partial sums behave like those of a geometric series.