In your role as a financial analyst, you are reviewing a long-term revenue model based on geometric growth. You need to recall the mathematical rule that determines if the total cumulative sum will grow indefinitely. By definition, an infinite geometric series is considered divergent—meaning its partial sums grow without bound and no finite sum exists—when its common ratio, $r$, satisfies which of the following conditions?

Classifying Infinite Growth in Financial Models

Suppose a sustainability coordinator is modeling a long-term resource usage pattern where the growth follows a geometric sequence. If the absolute value of the common ratio, $r$, satisfies the condition $|r| \geq 1$, the resulting infinite series is classified as divergent.

As a business operations analyst, you need to identify patterns of infinite growth in your models. Match each mathematical term or behavior associated with a divergent infinite geometric series with its corresponding description or characteristic.

As a financial auditor reviewing long-term cost projections, you identify a pattern of expenses that forms an infinite geometric series. If the absolute value of the common ratio is 1 or greater ($|r| \geq 1$), the partial sums will grow without bound. Because it has no finite total sum, this type of series is classified as a ____ series.