A corporate sustainability coordinator is modeling the long-term energy savings from a new high-efficiency lighting system. The savings each year follow a geometric series with an infinite number of terms. In order for the total cumulative savings to approach a finite, fixed value (meaning the series is convergent), which condition must the common ratio $r$ of the series satisfy?

A manufacturing company is modeling the total lifetime productivity of a specific machine. The productivity in each subsequent year is a constant fraction of the previous year's output, forming a geometric series with an infinite number of terms. True or False: For the total cumulative productivity to be convergent (meaning it approaches a fixed, finite limit), the absolute value of the common ratio $r$ must be less than ${}1$.

A sustainability consultant is modeling the long-term waste reduction of a facility that improves its efficiency by a constant percentage each year. Match the following mathematical properties of a convergent infinite geometric series with their correct descriptions.

Corporate Cost-Reduction Modeling

A financial analyst is modeling the total lifetime maintenance costs of a new manufacturing plant. The additional cost incurred each year shrinks progressively, forming an infinite geometric series. For the total cumulative maintenance cost to approach a fixed limiting value (meaning the series is convergent), the absolute value of the common ratio $r$ must be less than ____.

Condition for Convergence in Cost Projections

A corporate sustainability analyst is modeling the long-term, declining carbon emissions of a factory. The annual emissions decrease each year, forming an infinite geometric series. To ensure the cumulative lifetime emissions approach a finite, stable limit, the analyst needs to confirm that the series is convergent. Arrange the steps below in the correct order to mathematically evaluate whether this model represents a convergent infinite geometric series.