Example: The Multiplier Effect as an Infinite Geometric Series

In financial planning and engineering, we often calculate the total value of a process that decreases by a constant percentage over time. Match each component of the infinite geometric series model to its correct description.

In a corporate financial model, a manager is estimating the total long-term value of a service contract that generates decreasing annual returns. If the initial revenue in the first year is $a_1$ and each subsequent year's revenue is a fraction $r$ of the previous year, which of the following formulas correctly represents the total infinite sum $S$, assuming that $|r| < 1$?

In professional financial forecasting, the formula $S = \frac{a_1}{1 - r}$ can be used to calculate the total value of a geometric series with an infinite number of terms only if the common ratio $r$ satisfies the condition $|r| < 1$.

Formula for the Sum of an Infinite Geometric Series

In professional modeling, analysts often derive long-term results by analyzing the limit of a geometric process. Arrange the following steps in the correct logical sequence used to derive the formula for the sum of an infinite geometric series from the partial-sum formula, assuming $|r| < 1$.

In business forecasting, a marketing analyst is projecting the total long-term impact of a customer referral campaign. Each week, the number of new referrals declines by a constant ratio $r$. To calculate the total projected referrals over an infinite timeframe using the formula $S = \frac{a_1}{1 - r}$, the campaign must eventually stabilize. If the absolute value of the common ratio is greater than or equal to 1 ($|r| \ge 1$), the geometric series is ____, meaning a finite sum cannot be calculated.

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