Derive Definition of Exponential Function (Power Series) from Compound Interest

This example demonstrates how the formula for compound interest can be used to derive the power series definition of the exponential function. The power series of the exponential function is shown below.

From a high-level perspective, compound interest represents an iterative approach to modeling exponential growth and the exponential function is a natural limit to how fast something can “continuously” grow.

Start with the compound interest formula shown below.

The variables and their meanings are summarized below.

Then, let’s look at some examples of the formula. Specifically, we are interested in what happens as the frequency of investment, represented by the variable , increases.

Given an interest rate of , a principal amount of , total time of and a yearly investment strategy per year the formula calculates the accumulated amount shown below.

Next, given the same interest rate, principal amount and time elapsed, let’s calculate the accumulated amount if we switch quarterly investment strategy per year.

Finally, given the same interest rate, principal amount and time elapsed, let’s calculate the value for a monthly investment strategy per year.

These examples demonstrate that as the frequency of investment goes up so does the accumulated amount. The question we are trying to answer is “what happens as the frequency of the growth rate continues to increase?” Is there a natural limit, or does it grow in an unbounded manner?

We can pose this question mathematically by taking the limit of the function as goes to infinity. However, finding the limit at this stage poses some challenges and so we are first going to transform the function to be more simple and abstract.

Substitute the variable into the expression, where represents the total number of times the interest is applied and is given by . This changes the function to the equivalent form:

Then, we are going to make two observations. First, observe that scales the output (growth curve) of the function vertically. Second, observe that scales the input of the function horizontally. Since we are interested in modeling the growth curve (shape) of the function and because it makes the math cleaner we can set and .

To finish the transformation, from now on, let’s consider to represent the total number of times interest is applied and replace with the more generic variable .

Take the limit of the function as approaches infinity.

There are two reasonable approaches to taking the limit of this function. The first, shown on this page, is to expand the product as a series and look for patterns. The second, shown on another page, is to approximate a value for the limit.

Expand the first three cases of the limit.

This is tedious to do by hand and instead, we can use the binomial expansion, substituting and into each case of the expansion. This is shown below and aligned so that the powers of line up.

The math becomes pretty hairy at this point, and while we won’t prove anything rigorously, hopefully, you will be convinced that as the coefficients approach known values. Take, for example, the expansion of the expressions of where is equal to , and shown below. Clear patterns start to appear in the coefficients associated with each power. For example, the coefficient associated with approaches the fraction .

The generalized pattern that emerges as approaches infinity is the coefficient in front of the -th power of approaches over the factorial of . This pattern leaves us with the infinite power series below which represents the definition of the exponential function. This is the same definition that can be derived using a Taylor series^[1].

We can give this function the abbreviated name and we have finished the derivation of the power series definition of the exponential function.

If desired, the scalar values of the principal and interest rate can be substituted back into the formula to recover the ability to scale the function vertically and horizontally. This formula is the same as the population growth formula.

1. Derive Definition of Exponential Function (Taylor Series) Example
   https://wumbo.net/examples/derive-exponential-function-taylor-series/