Consider the infinite geometric series

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

S=a+ar+ar^{2}+ar^{3}+\text{···}

. (i) State the condition on the common ratio

r

under which the above series converges. (ii) Using the behaviour of the term

r^{n}

n\to \infty

, explain clearly why the series does not converge when

|r|\ge 1

. (iii) Take

a=3

and

r=2

and write the first four partial sums of the series. Comment on their trend and relate it to your explanation in part (ii).