The sum of an infinite geometric series exists only if the condition on the common ratio r is

∣r∣<1

∣r∣≤1

∣r∣>1

∣r∣≥1

Explanation

An infinite geometric series converges (has a finite sum) only if the absolute value of the common ratio, |r|, is less than 1. If |r| ≥ 1, the series diverges, meaning it does not approach a finite sum.