Mathematics

Problem

Let

a

be the first term and

r\,(r\neq 1)

the common ratio of a geometric progression. a. (i) Show that the sum of the first

n

terms is

S_n = a\dfrac{1-r^{\,n}}{1-r}

. b. (ii) Evaluate

\displaystyle \lim_{n\to\infty} S_n

when

|r|<1

and hence obtain a formula for the sum to infinity

S_{\infty}

. c. (iii) Explain why the infinite series cannot possess a finite sum when

|r|\ge 1