Name: compound interest and series growth
Uploaded: 2026-06-08T09:38:55.180Z
Duration: 1 min 19 s
Channel: JoVE

Consider an infinite series. To understand whether this infinite series converges to a finite value or grows endlessly, one uses the Ratio Test. This test examines the limit of the absolute value of the ratio between consecutive terms, called L.If L is less than one, the series converges, meaning its sum reaches a finite number. If L is greater than one or equals infinity, the series diverges, meaning the sum grows without bound.When L equals one, the test gives no clear result, and other methods are needed.For example, take a principal amount invested that grows each year at a 4% compound interest rate. This growth creates a geometric sequence of terms, where each new term is 1.04 times the previous one.Because this value is greater than one, each term increases with each compounding period, and the total sum grows without bound.The series diverges, showing that the accumulated sum increases without limit, and its infinite total cannot be calculated.

the ratio test

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&#x2019;s balance becomes 1.04 times the previous year&#x2019;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.

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compound interest and series growth

Compound interest grows an investment by adding interest to both the original amount and the interest already earned. With a 4% annual compound interest rate, each year’s balance becomes 1.04 times the previous year’s balance. That pattern creates a geometric sequence, with 1.04 as the common ratio or repeated multiplier.

Mathematicians use the Ratio Test to check whether an infinite series has a finite sum or keeps growing without limit. The test compares consecutive terms in the series and looks at the ratio, called L. If L is less than one, the series converges to a finite sum. If L is greater than one or infinite, the series diverges. If L equals one, the test does not give a clear answer.

For the 4% compound interest example, the common ratio is 1.04. Because this value is greater than one, the geometric series diverges. That means the investment balance keeps increasing without bound as time goes on, so its total value over an infinite timeline cannot be calculated. This idealized model assumes infinite time and no withdrawals, but it still shows the accelerating growth of fixed-rate compound interest.

Compound Interest and Series Growth

introduction to sequences

Zeno’s Paradox and Geometric Sequences

Zeno’s paradox shows how geometric sequences can describe motion that keeps getting smaller. In the example, a man walks toward a door but only covers half of the remaining distance each time. That creates a sequence of steps: one-half, one-quarter, one-eighth, and so on. This pattern is a geometric sequence because each term is half of the previous term. It can be written as a_n = 1/2^n, where n is the step number starting at 1. The terms shrink in a regular way, even though the sequence...

convergence of sequences

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introduction to infinite series

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partial sums and series convergence

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the integral test

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comparison tests

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Comparison tests help determine whether an infinite series of positive terms converges or diverges. Instead of finding the sum directly, these methods compare the series with another series whose behavior is already known. That comparison gives a clear answer about whether the total stays finite or grows without bound. The direct comparison test works when each term of an unknown positive series can be matched to a second series term by term. If the unknown terms are less than or equal to the...

alternating series and absolute convergence

alternating-series-and-absolute-convergence

Damped Springs and Alternating Series

Damped spring motion can be modeled with alternating series. A mass on a vertical spring moves above and below a central equilibrium point. In an ideal spring, the motion would keep the same amplitude. In a damped spring, resistive forces such as air resistance or internal friction make each swing smaller over time. This motion is often written as a sinusoidal function with an exponential decay factor. The sine term describes the repeating back-and-forth motion. The decay factor lowers the...

Compound interest grows an investment by adding interest to both the original amount and the interest already earned. With a 4% annual compound interest rate, each year’s balance becomes 1.04 times the previous year’s balance. That pattern creates a geometric sequence, with 1.04 as the common ratio or repeated multiplier. Mathematicians use the Ratio Test to check whether an infinite series has a finite sum or keeps growing without limit. The test compares consecutive terms in the series and...

power series and their properties

Power Series as Function Approximations

Power series can represent a function as an infinite sum of terms with powers of a variable. The terms are usually written around a central value, called the center. That center is shown as a, so the powers become (x − a)n, where n is a non-negative integer. The coefficients in the series control the behavior of the function. When the center is zero, the series becomes a power expansion in x. It then looks much like an infinite polynomial. This form can model complex functions with high...

interval and radius of convergence

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Power series converge only for certain input values, so it is important to find the range where they work. The radius of convergence gives the distance from the center where the series still converges. The interval of convergence shows the real x-values that fit inside that range. A general power series can be written as a sum of terms of the form c_n(x - a)^n, where a is the center. To find the radius of convergence, the ratio test is often used. It compares consecutive terms by taking the...

bessel function of order zero

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taylor series

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Taylor series match a smooth function near a chosen center by using its derivatives. The center is the point where the function is analyzed, and the goal is to build a power series that follows the function’s local behavior. In this expansion, each coefficient is chosen so the series and the function have the same value and the same derivatives at the center. Start with a power series centered at x = a. It has the form Σn=0∞ cn(x − a)n. When x = a, every term with (x − a) becomes zero, so only...

convergence of taylor series

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Maclaurin Series for Cosine Motion

Maclaurin series turn a smooth function into an infinite polynomial centered at zero. A Maclaurin series is a Taylor series with the center at zero, so every derivative is taken at that point. That setup often makes the coefficients easier to find. These series are useful in applied mathematics and physics because they replace complicated functions with polynomials that are easier to analyze and compute. One standard example is the cosine function. Its Maclaurin series is cos x = 1 - x^2/2! +...

binomial series

Power Series for Fractional Exponents

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key terms and concepts

See how compound interest creates a geometric sequence and how the Ratio Test shows when the related series converges, diverges, or is inconclusive.

Compound Interest and Series Growth | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Compound Interest and Series Growth

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