When I started trying to think about applications of infinite series outside of mathematics, my mind quickly went to the business school where I spend the majority of my time when I am not in math classes. Early in introductory finance or economics classes, students are introduced to the idea of the Time Value of Money. The idea of the Time Value of Money is that money that you don’t have yet, but are receiving down the road, is worth less than that same amount of money if you already have it in your hand.

 

Intuitively, this makes sense. If I were to offer you $100 right now or $100 one year from now, you would surely prefer to take the money now. But what if I were to offer you $100 right now or $110 a year from now? Now it starts getting tricky and that is where the Time Value of Money comes in. The value which some amount of money in the future is worth right now is called its Present Value. Present Value is computed by determining how much additional value could have been earned from interest if you had the same amount of money right now and invested it. For example, if the interest rate were 10%, and you invested $100 today, in one year it would be worth $110. Thus, the Present Value of $110 is $100.

In general the formula to compute Present Value is

For example, if the interest rate is 10%, the present value of $100 in 2 years would be:

The key here is that by not having the money for two years you miss out on the ability to invest the money in year one, and also the ability to invest the original money plus year one’s interest in year two.

The Time Value of Money can also be used to compute the Present Value of multiple payments of money in the future. Imagine collecting $100 on your birthday for the next five years. This would be called a five year annuity of $100. You could compute the present value of the annuity by summing the present values of each of those five payments. That computation would be expressed by the following series:

As is often the case, the computation is pretty simple when the scale is small, but quickly gets complicated as the scale (in this case number of years for the annuity) increases. Consider an annuity that were to recur forever. In finance and economics, this is called a perpetuity. A perpetuity is expressed by the following series:

Therefore, a perpetuity is an example of an infinite geometric series. A good examples of this is a consol which is a special kind of bond issued by the British government. Consoles promise to pay the bondholder interest payments forever, and these infinite future payments would be expressed through an infinite series as shown above. Although consoles, or any kind of perpetuity, represent the sum of an infinite amount of payments, it is crucial that they are able to be expressed by a single, finite present value, or it would be impossible to buy, sell, exchange, and appraise them. Thus, the fact that the series converges is of utmost importance. By slightly rearranging the series, it can be expressed as

Thus, the geometric series has a common ratio of and since the common ratio is less than 1, the series converges. As an example, consider a perpetuity of $100 per year with 10% interest.

The sum is expressed as the first term divided by 1 minus the common ratio:

And in general the present value of a perpetuity is simply expressed as .

Another example of infinite series can be found in multimedia. Jean-Baptiste Joseph Fourier (1768-1830) made the important discovery that nearly any periodic function can be expressed as an infinite series of sin and cosine functions. This particular type of infinite series is known as a Fourier series. Fourier had been trying to solve the heat equation, which was a partial differential equation. At the time, it had only been possible to solve the equation in the specific instances where the heat source was either a sine or cosine wave. But Fourier was able to show that an infinite series of sine and cosine waves of various lengths and amplitudes would be able to reproduce nearly any periodic function, which allowed him to solve the equation in the general sense. Today, Fourier series are common and are used frequently in engineering, vibration analysis, harmonic analysis, audio, and quantum mechanics.

However, a simple application that many of us are very familiar with is compression in multimedia. This application also highlights an interesting and useful characteristic of these infinite series. Think of your favorite song, and then imagine the sound wave which produces it. We know that there is likely a Fourier series which can express the sound wave extremely well. Conveniently, when a Fourier series converges, it will converge very quickly, and after only a few terms in the series, it will already be extremely similar to the original sound wave. In fact, after only a few terms in the series, the difference between the terms from the Fourier series and the original sound wave will be so minimal that the changes will not be able to be detected when you listen. This allows an extremely good approximation of the original sound wave to be created and all the remaining terms of the infinite Fourier series to be truncated. The result is that your favorite song can be compressed and saved as a much smaller file. A very similar application allows photos to be compressed and expressed as truncated Fourier series. So next time you download an .mp3 or .jpeg, you likely have Jean-Baptiste Joseph Fourier and infinite series to thank for how quickly the file downloads and how little space it takes up.