Mathematics

Problem

Consider the infinite geometric progression

a,ar,ar^{2},ar^{3},\dots

where

a\neq 0

. a. (i) Assuming that

|r|<1

, derive an expression for the sum to infinity

S_{\infty}

in terms of

a

and

r

. b. (ii) Conversely, prove that if the series possesses a finite sum

S_{\infty}

, then necessarily

|r|<1

. c. (iii) Combine the above results to establish that the condition

|r|<1

is both necessary and sufficient for an infinite geometric progression to converge.