To determine the sum of an infinite geometric series, we need to consider the common ratio [tex]\( r \)[/tex] and the properties of geometric series.
An infinite geometric series is given by:
[tex]\[ S = a + ar + ar^2 + ar^3 + \cdots \][/tex]
where [tex]\( a \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
For an infinite geometric series to have a sum, it must converge. The criteria for convergence is:
[tex]\[ |r| < 1 \][/tex]
When [tex]\( |r| < 1 \)[/tex], the sum [tex]\( S \)[/tex] of the series can be determined using the formula:
[tex]\[ S = \frac{a}{1 - r} \][/tex]
However, if [tex]\( |r| > 1 \)[/tex], the series does not converge. In other words, the values of the terms keep increasing or decreasing without bound, and hence the series diverges.
When the series diverges, it means that it does not sum to a finite value and thus:
[tex]\[ \text{The sum of the infinite geometric series cannot be determined when } |r| > 1. \][/tex]
Therefore, the correct answer is:
A. cannot be determined
