Infinitely Marvellous: The Beauty of Infinite Series in Math and Science

Infinity boggles the mind, even for mathematicians. For thousands of years, the idea of the infinite has captured the imaginations of eminent mathematicians like Euler, Cantor, Reimann and Ramanujan. Part of the charm of infinite series is that they are among the most powerful concepts in the whole of math, capable of unifying seemingly disparate ideas across vast mathematical terrains. In this post, I like to give you a flavor of this unifying power of infinite series, using simple examples that will not tax our (finite) brains.

I will begin with a fun problem about two cyclists and a buzzing fly. Then, I will relate this problem with a true story of a great mathematician who solved this puzzle in his head within seconds. After that, I will show how you can solve it too, albeit with a bit of help from an infinite series known as geometric progression (GP). Finally, I will end this post with an application of GP to the field of finance, presenting an equation is is simple, elegant, and widely used. Ready? Let’s get going.

A Fun Problem

Two bicyclists start at opposite ends of a road 20 miles long. Each cyclist travels toward the other at 10 miles per hour. When they begin, a fly sitting on the front wheel of one of the bikes takes off and races at 15 miles per hour toward the other bike. As soon as it gets there, it instantly turns around and zips back toward the first bike, then back to the second, and so on. It keeps flying back and forth until it’s finally squished between their front tires when the bikes collide. How far did the fly travel, in total, before it was squished?

This is the kind of problem that either gives you a migraine or a chance to prove you’re a math genius. Give it a go and see if you can solve the problem. Then I’ll tell you the promised story and the solution.

Solution to the fly problem

To solve the problem, we must figure out where the fly traveling at 15 miles an hour first meets the bicycle approaching it at 10 miles an hour. Because their speeds form the ratio 15:10, or 3:2, they meet when the fly has travelled 3/(3+2) = 3/5th of the initial 20-mile separation, or 12 miles. With a bit of effort, you could deduce that each leg of the fly’s back-and-forth journey is one-fifth as long as the previous leg. So, by the second leg, the fly would have flown 12 + 12 x 1/5 miles. And by the third leg, it would have flown by an addition 12 x (1/5)^2 miles and so on. In other words, to find the answer, add to the first number (12) add successive terms, each equal to 12 times 1/5 raised to ascending powers ad infinitum. You can do this easily on Excel and come up with an approximate solution of 15 miles (approximate because there are only finite number of cells in Excel).

The fly problem was posed to the great Hungarian mathematician, John von Neumann (1903-57), who is regarded as the father of the modern computer, a pioneer of game theory and above all, a genius who possessed a lightning quick mind. The story goes that when von Neumann heard the puzzle, he instantly replied, “15 miles” , to the disappointment of his questioner who said, “Oh, you saw the trick.”

“What trick?” said von Neumann. “I just summed the infinite series.” And he was right. Von Neumann knew that the fly problem involves what mathematicians call a geometric series, also known as geometric progression (GP) whose terms sum to infinity as follows:

S = a + ar + ar2 + ar3 + …

where S stands for sum, r is a ratio and a is what’s called the leading term of a GP. If the ratio lies between -1 and 1, as it does in our problem, S simplifies to the elegant formula:

\displaystyle S = \frac {a}{1-r}

Since a = 12, and r = 1/5, these numbers imply a solution of 15 miles.

The Tentacles of Infinite Series

Now back to the bigger picture: how do infinite series like the GP help to connect various parts of math? To see this, we need to enlarge our viewpoint about formulas like

1 + r + r2 + r3 + … = 1/(1-r),

which is the same as the GP formula except that a is now equal to 1. And instead of thinking of r as a specific number like 1/5 or -1/5, think of it as a variable, a number that can change. Then, something amazing happens: the formula now expresses a kind of mathematical magic, as if lead could be turned into gold. The more general formula asserts that a given function of r (here, 1 over 1-r) can be turned into a combination of simple powers of r, like r^2 and r^3 and so on.

What’s fantastic is that the same is true for an enormous number of other functions that appear virtually everywhere in science and engineering, things like sines and cosines, logarithms and exponentials. The result is a family of infinite power series that are like a beefed-up version of a GP where the coefficients may now also change.

When mathematicians made these conversions, they noticed startling coincidences. Here are a few examples (don’t worry about the expressions come from; just look at their appearance):

\displaystyle \cos x = \frac{x^2}{2!} + \frac{x^4}{4!}- \frac{x^6}{6!} + ...

\displaystyle \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...

\displaystyle e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...

Besides all the exultant exclamation points (which stand for factorials; 4! means 4 x 3 x 2 x 1, for example), notice that the series of \displaystyle e^x comes tantalizingly close to being a mashup of the two formulas above it. If only the alternation of positive and negative signs in \displaystyle \cos x and \displaystyle \sin x could somehow harmonize with the all-positive signs of \displaystyle e^2 , everything would match up.

The Most Beautiful Equation in Math

That coincidence, and that kind of wishful thinking, led the great 18th century mathematician Leonard Euler to the discovery of one of the most marvelous formulas in the history of mathematics:

\displaystyle e^{ix} = \cos x + i \sin x,

where i is the imaginary number defined as \displaystyle i = \sqrt{-1} .

(For a lucid derivation of Euler’s formula, have a look at the video at the end of this post)

Euler’s formula says something remarkable; it assets that sines and cosines, the embodiment of cycles and waves, are secret relatives of the exponential function, the embodiment of growth and decay — but only when we consider raising e to an imaginary power.

Euler’s formula, spawned directly by infinite series, is universally regarded as the most beautiful equation in math. Not only that, it is now indispensable in many fields of science and engineering, including electrical and sound engineering, semiconductor research, quantum mechanics and indeed, all technical disciplines concerned with waves and cycles. Meanwhile, our good old GP also makes a starring role in the field of finance, which I now turn to.

An Infinite Series on Wall Street

An important question in finance is how to ascertain the intrinsic value of risky securities like stocks. In other words, how much is a stock worth based on its future cash flows, discounted by a factor that reflects the riskiness of those cash flows?

For stocks, the cash flows available to investors are the company’s dividends declared out of their earnings. Past dividends are already paid, so what matters for the intrinsic price of a stock is future dividends. A model known as the Constant Dividend Growth Model or CGDM is widely used by investor professionals to calculate the intrinsic price of a stock. It assumes that the company is a going concern, and that its dividend per share will grow at a long-term steady state rate (let’s call it g). Given these assumptions, the intrinsic price per share can be written as:

\displaystyle P = \frac{D_1}{(1+r)} + \frac {D_1(1+g)}{(1+r)^2} + .\frac{D_2(1+g)^2}{(1+r)^3} + ...

where P stands for the intrinsic price, D_1 is the expected dividend per share in the next period, D_2 is the expected dividend per share in the next next period and so forth. Finally, r now stands for the discount rate which reflects the riskiness of the firm’s expected dividends. The riskier the stock, the higher the discount rate. Otherwise, one risk paying too much (the business of estimating the discount rate deserves another blog, which I will omit for now).

Multiply both left- and right-hand side of the above equation by (1+r). This gives:

\displaystyle P(1+r) = D_1 + \frac {D_1(1+g)}{(1+r)} + .\frac{D_2(1+g)^2}{(1+r)^2} + ...

The right-hand side is now a geometric progression with

\displaystyle a = D_1 \hspace{0.2cm} and \hspace{0.2cm} \frac {1+g}{1+r} = r .

Using the solution to the infinite sum then gives:

\displaystyle P(1+r) = \frac{D_1}{1- \bigg(\frac{1+g}{1+r} \bigg)   }  \hspace{0.2cm} or

\displaystyle P(1+r) = \frac{D_1(1+r)}{(1+r)-(1+g)}

The two (1+r) terms cancel out, leaving:

\displaystyle P = \frac{D_1}{r-g}

This elegant equation gives the intrinsic price of a stock as a function of the next period dividend per share, the discount rate and the constant growth rate of dividends per share going forward. This price will be positive if the discount rate is greater than the dividend growth rate, a reasonable assumption for the long run (remember that the CGDM is a steady state model, not one to forecast next week or next month’s stock price).

We can flip the equation around to get an expression for the discount rate which can be interpreted as the expected long-run rate of return the stock yields given the assumptions.

\displaystyle r = \frac{D_1}{P} + g

From this expression, we have the intuitive result that stocks are profitable, pay good dividends and have high dividend growth rates will tend to have high expected returns. This prediction has been borne out in empirical research, a testimony to the predictive power of infinite series applied to finance.

Video: Euler’s Formula from Sine and Cosine Infinite Series

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