Consider the convergent series k=1∑∞ak with sum S. Which of the following best explains why the difference ∣S−Sn∣ 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 ak approach 0 as k→∞.
c) Because the sequence of partial sums {Sn} approaches a finite limit as n→∞.
d) Because the successive ratios akak+1 are eventually less than 1, so the partial sums behave like those of a geometric series.