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

This example demonstrates how the concept of compound interest can be used to derive the power series definition of the exponential function. Note, this gives the same definition as deriving the exponential function using a Taylor Series. The power series definition 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. The figure below suggestively illustrates how the two are connected. Note how for compound interest, the growth at each step “compounds”, meaning that it is proprotional to the current value of the investment.

This figure illustrates the relationship between compound interest and exponential growth.
Figure 1: Compound Interest and Exponential Growth

Steps

  1. To start, we are going to pose question that will help us discover the exponential function: “Given a growth rate (compounding interest) and an interval of time. What is the largest amount of growth possible?” To answer this question we are going to use a variation of the compound interest formula shown below that is written in terms of total time and total number of times the interest is applied.

    Expression Description
    Accumulated amount
    Principle amount
    Rate of interest
    A time interval representing the total elapsed time
    Total number of times interest is applied in the time interval.

    First, let’s look at a real-world example to show how compound interest models exponential growth and how the variable influences the growth. Then in the next step we will transform our formula to be more abstract. All of the examples below use an principle amount of and an interest rate of . The difference we are interested in between the formulas is the frequency in which the interest is applied.

    For example, given an interest rate of , a principle amount of , total time of and a yearly investment strategy which sets , let’s calculate the accumulated amount.

    Yearly Compound Interest Example
    Figure 1: Compound Interest Yearly Example

    Now let’s look at what happens if we switch to a quarterly investment strategy. This means the interest will be compounded times per year so over the course of years, . Note, the amount of interest compounded gets scaled appropriately. In this case, each time the interest is compounded by .

    This figure is a graph of a quarterly compound interest example.
    Figure 2: Compound Interest Quarterly Example

    As we can see from these two examples, as increases, so does the accumulated amount of money. So, what happens if we chop the interval into finer and finer increments until we are continously compounding the interest? In other words, what happens when approaches infinity. We can formally define this question using the limit notation in the equation below.

    As we will see, it turns out there is natural limit to continuous growth. This natural growth curve is defined by the exponential function.

  2. Modify the formula to be more abstract. Set the principle amount to and express as the variable using the two observations below.

    • The principle amount scales the function vertically, so we can generalize the formula so the principle amount is simply .

    • The interest rate scales the function horizontally and because it is a constant, the expression can be abstracted using one variable . This step isn’t required, we could proceed with in the equation, but it will make the math cleaner and later we can recover the notion of a growth constant as is the case in the formula for population growth.

    Putting these abstractions together with the limit notation gives us a definition for the exponential function, denoted using , shown below.

    However, we would prefer not to have a limit in the definition, so we need some way to evaluate the limit.

  3. There are two reasonable approaches to evaluate the limit. The first, outlined here, is to simply plug in values for and expand the expression and see what kind of pattern emerges. This gives us an infinite power series. The second, outlined in another example, uses algebra to manipulate the expression so that the limit can be given an approximate value, which leads to another definition of the exponential function.

    Let’s start by expanding the first three cases by hand. These are pretty straightforward, but as you can imagine, it gets pretty tedious.

    Case Expression

    A better approach is to use binomial expansion. Shown below are the first four cases of binomial expansion where and . The coeffecients highlighted in blue correspond to Pascal’s Triangle.

    Case Expression Binomial Expansion

    Then, align the expansion so that powers of match up. Now we can see if a trend emerges. The claim here, without going into a rigorous proof, is that the coeffecients for each power of approach a known constant.

    Case Binomial Expansion

    As the last line of the table claims, where , the coeffecients for each power of approach a known constant. The clearest example of this is the coeffecients which approach . The coeffecients of the next term of the series, where , approach the fraction . This is shown be the series below.

    Similarily, if we write out the series for the next power of , the coeffecients for the term of the expansion this series approaches .

    The key insight here is that a pattern emerges for the coeffecients of each of the powers of : , , , , , . The coeffecient for some will approach the value . This gives us the final form of the exponential function shown below, which is identical to the definition found by using a Taylor series to define function[1].