Question

Explain why \(A=P\left(1+\frac{r}{n}\right)^{n t}\) approximates \(A=P e^{r t}\) as \(n\) approaches positive infinity.

Short answer

The discrete compound interest formula approximates the continuously compounded interest formula because as the number of compounding periods \(n\) increases indefinitely or reaches infinity, the process of adding interest becomes continuous. This leads it to approximate Euler's number to the power of the product of the rate of interest and the time, hence approximating the formula for continuously compounded interest, \(A = P e^{rt}\).

Step-by-step solution

Understanding the given formulas

The expression \(A=P\left(1+\frac{r}{n}\right)^{n t}\) is known as the formula for discrete compound interest. Here, A is the final amount of money, P is the principal amount (the initial amount of money), r is the interest rate (in decimal form), n is the number of times that interest is compounded per unit t, and t is the time the money is invested for. As for \(A = P e^{rt}\), it represents continuously compounded interest.

Introducing the concept of limit

The concept of limit in calculus is a fundamental idea that the rest of calculus relies upon. It is the value that a function or sequence 'approaches' as the input or index approaches a certain value. In the given problem, we are interested in what happens when the compounding frequency \(n\) gets arbitrarily large (approaching infinity).

Recognizing Euler's Number

Euler's number, represented by \(e\), is a mathematical constant approximately equal to 2.71828. One of the key characteristics of this number, which makes it very important in calculus and its applications, is that the function \(e^x\) is its own derivative and its own integral.

Explaining the approximation

As \(n\) approaches infinity in the expression \(A=P\left(1+\frac{r}{n}\right)^{n t}\), the recurrence of compounding becomes increasingly higher. This essentially means that interest is being added an infinite number of times, however small, over the given period of time. This comes ever closer to being continuous, thus approximating \(e^{rt}\), the expression for continuous compounding.

Key concepts

Discrete Compound Interest

Discrete compound interest refers to the method of calculating interest on a principal sum where the interest is added at fixed intervals of time. The formula for discrete compound interest is given by: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:

• \(A\) = Accumulated Amount after time \(t\)
• \(P\) = Principal amount (initial investment)
• \(r\) = Annual interest rate (as a decimal)
• \(n\) = Number of compounding periods per year
• \(t\) = Number of years the money is invested for

Discrete compounding assumes that interest is applied in equal chunks throughout the year. For example, quarterly compounding would apply interest four times a year. As \(n\) increases, more frequent compounding occurs, allowing the invested sum to grow faster than with less frequent compounding. This concept underpins traditional savings and investment accounts, where interest might be compounded monthly or annually.

Continuously Compounded Interest

In the world of finance, continuous compounding is an idealized concept where interest is applied and accumulated instantly at every possible moment. This is represented by the formula: \[ A = P e^{rt} \]Where:

• \(A\) = Final amount after time \(t\)
• \(P\) = Initial principal amount
• \(e\) = Euler's number, approximately 2.71828
• \(r\) = Annual interest rate (as a decimal)
• \(t\) = Duration in years

Continuous compounding means that as time progresses, the money continually grows as the interest earned is instantly reinvested. It is a theoretical concept describing a limit where compounding periods occur at infinitesimally small intervals—truly every moment—with interest being added without break. This represents the maximum possible growth of an investment from compounding alone.

Euler's Number

Euler's Number, denoted as \(e\), is a fundamental constant in mathematics approximately equal to 2.71828. This number arises naturally in the process of continuous compounding and appears in various areas of mathematics. One of the key properties of \(e\) is that it provides the base of natural logarithms, which have widespread applications in calculus, growth processes, and differential equations.

Euler's Number is pivotal because it defines an exponential function that grows proportionally and continuously concerning its current value. That is to say:

• The derivative \(\frac{d}{dx}e^x = e^x\)
• The integral \(\int e^x \,dx = e^x + C\)
• \(e^x\) describes natural exponential growth

In finance and economics, \(e\) is the cornerstone for calculating continuously compounded interest, highlighting natural growth processes.

Limit in Calculus

In calculus, the concept of a limit is the foundation for derivatives and integrals. A limit helps us understand the behavior of a function as its input approaches a particular value. It can be thought of as the function's tendency when approaching a certain point or boundary. For example, to better understand continuously compounded interest, we consider the limit of discrete compound interest: \[ \lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^n = e^r \]

This shows how increasing the frequency of compounding (larger \(n\)) tends the compound interest formula towards the natural exponential base \(e\). When \(n\) approaches infinity, the process of interest compounding becomes continuous, converging to the expression for continuously compounded interest. The idea of limits connects calculating interest to broader mathematical concepts, offering deep insights into growth and change.