The problem asks for the condition under which the sum to infinity of a geometric progression exists. The options are various inequalities involving $r$, the common ratio of the geometric progression.

1. Problem Description

The problem asks for the condition under which the sum to infinity of a geometric progression exists. The options are various inequalities involving $r$, the common ratio of the geometric progression.

2. Solution Steps

The sum to infinity of a geometric progression exists if and only if the absolute value of the common ratio, $|r|$, is less than

1. In other words, $-1 < r < 1$. If $|r| \ge 1$, the terms of the geometric progression do not approach zero, and the sum to infinity does not converge.

The sum to infinity of a geometric progression is given by:

$S = \frac{a}{1 - r}$,

where $a$ is the first term and $r$ is the common ratio. This formula is valid only when $|r| < 1$.

The options are:

a) $r = 1$ - incorrect, because $|1| = 1$, so $|r| \ge 1$.

b) $|r| \ge 1$ - incorrect, as this condition implies the sum to infinity does not exist.

c) $r = 2$ - incorrect, because $|2| = 2$, so $|r| \ge 1$.

d) $|r| < 1$ - correct.

e) $r > 1$ - incorrect, because if $r>1$, $|r| = r > 1$, so $|r| \ge 1$.

3. Final Answer

d. absolute r < 1