Which of the following correctly explains why the infinite series does NOT possess a…
MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Which of the following correctly explains why the infinite series 5+10+20+40+… does NOT possess a finite sum?
(a) Applying S=1−ra gives S=1−25=−5; a negative sum for positive terms is impossible, so the series has no finite positive sum.
(b) Its common ratio is 2 and, since ∣2∣>1, the terms grow without bound, so the series diverges.
(c) The difference between successive terms is not constant, hence no finite sum exists.
(d) Because the series never alternates in sign, the terms cannot cancel, leading to divergence.