Let be the first term and the common ratio of a geometric progression.

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Let

a

be the first term and

r\,(r\neq 1)

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

n

terms is

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

. (ii) Evaluate

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

when

|r|<1

and hence obtain a formula for the sum to infinity

S_{\infty}

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

|r|\ge 1