The Ratio Test

Compound interest describes how an investment grows when interest is added not only to the initial amount but also to previously earned interest. With a 4% annual compound interest rate, each year’s balance becomes 1.04 times the previous year’s amount. This growth forms a geometric sequence, where the common ratio of 1.04 represents the consistent multiplier applied to each term.

To analyze whether an infinite sum of such terms reaches a finite value or grows indefinitely, mathematicians use the Ratio Test. This test examines the ratio between consecutive terms in the series. If the ratio, called L, is less than one, the series converges to a finite sum. If L is greater than one or infinite, the series diverges, meaning its total increases without bound. If L equals one, the test is inconclusive.

In the case of the 4% interest rate, the common ratio is 1.04. Since this value is greater than one, the corresponding geometric series diverges. This means that as time progresses, the investment balance grows without limit, and its total value over an infinite timeline cannot be calculated.

The divergence of the series reflects the nature of compound interest, which is characterized by continuous growth without a cap. While this idealized model assumes infinite time and no withdrawals, it highlights the accelerating nature of exponential financial growth under fixed-rate compounding.