# How Infinite Series Reveal the Unity of Mathematics

## Introduction

For sheer brilliance, it was hard to beat John von Neumann. An architect of the modern computer and inventor of game theory, von Neumann was legendary, above all, for his lightning-fast mental calculations.

The story goes that one day somebody challenged him with a puzzle. 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?

It sounds hard. The fly’s back-and-forth journey consists of infinitely many parts, each shorter than the one preceding it. Adding them up seems like a daunting task.

But the problem becomes easy if you think about the bicyclists, not the fly. On a road that’s 20 miles long, two cyclists approaching each other at 10 miles per hour will meet in the middle after 1 hour. And during that hour, no matter what path the fly takes, it must have traveled 15 miles, since it was going 15 miles an hour.

When von Neumann heard the puzzle, he instantly replied, “15 miles.” His disappointed questioner said, “Oh, you saw the trick.” “What trick?” said von Neumann. “I just summed the infinite series.”

Infinite series — the sum of infinitely many numbers, variables or functions that follow a certain rule — are bit players in the great drama of calculus. While derivatives and integrals rightly steal the show, infinite series modestly stand off to the side. When they do make an appearance it’s near the end of the course, as everyone’s dragging themselves across the finish line.

So why study them? Infinite series are helpful for finding approximate solutions to difficult problems, and for illustrating subtle points of mathematical rigor. But unless you’re an aspiring scientist, that’s all a big yawn. Plus, infinite series are often presented without any real-world applications. The few that do appear — annuities, mortgages, the design of chemotherapy regimens — can seem remote to a teenage audience.

The most compelling reason for learning about infinite series (or so I tell my students) is that they’re stunning connectors. They reveal ties between different areas of mathematics, unexpected links between everything that came before. It’s only when you get to this part of calculus that the true structure of math — all of math — finally starts to emerge.

Before I explain, let’s look at another puzzle involving an infinite series. Solving it step by step will clarify how von Neumann solved the fly problem, and it will set the stage for thinking about infinite series more broadly.

Suppose you want to buy a fancy hat from a street vendor. He’s asking $24. “How about $12?” you say. “Let’s split the difference,” he replies, “$18.”

Often that settles it. Splitting the difference seems reasonable, but not for you, because you’ve read the same negotiation manual, “The Art of Infinite Haggling.” You counter with your own offer to split the difference, except now it’s between $12 and the last number on the table, $18. “So how about it?” you say, “$15 and it’s a deal.” “Oh no, my friend, let’s split the difference again, $16.50,” says the vendor. This goes on ad absurdum until you converge on the same price. What is that ultimate price?

The answer is the sum of an infinite series. To see what it is, observe that the successive offers follow an orderly pattern:

24 | his asking price |

12 = 24 − 12 | your first offer |

18 = 24 − 12 + 6 | splitting the difference between 12 and 24 |

15 = 24 − 12 + 6 − 3 | splitting it between 12 and 18 |

The key is that the numbers on the left side of the equal sign are built up systematically from the ever-lengthening series of numbers on the right. Each number appearing in the sequence (24, −12, 6, −3…) is half the number that precedes it, but with the opposite sign. So in the limit, the price *P* that you and the vendor will agree to is

*P* = 24 – 12 + 6 – 3 + …

where the three dots mean the series continues forever.

Rather than trying to wrap our minds around such an infinitely long expression, we can perform a cunning trick that makes the problem easy. It allows us to cancel out that bewilderingly infinite collection of terms, leaving us with something much simpler to calculate.

Specifically, let’s double *P*. That would also double all the numbers on the right. Thus,

2*P* = 48 – 24 + 12 – 6 + ….

How does this help? Observe that the infinite chain of terms in 2*P* is almost the same as that in *P* itself, except that we have a new leading number (48), and all the plus and minus signs for our original numbers are reversed. So if we add the series for *P* to the series for 2*P*, the 24s and the 12s and everything else will cancel out in pairs, except for the 48, which has no counterpart to cancel it. So 2*P* + *P*= 48, meaning 3*P* = 48 and therefore

*P* = $16.

That’s what you’d pay for the hat after haggling forever.

The problem of the fly and the two bicycles follows a similar mathematical pattern. 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. Von Neumann would have found it child’s play to sum the resulting “geometric series,” the special kind of series we’ve been considering, in which all consecutive terms have the same ratio. For the fly problem, that ratio is $latex\frac{1}{5}$. For the haggling problem, it’s $latex-\frac{1}{2}$.

In general, any geometric series *S* has the form

*S* = *a* + *ar* + *ar*^{2} + *ar*^{3} + …

where *r* is the ratio and *a* is what’s called the leading term. If the ratio *r* lies between −1 and 1, as it did in our two problems, the trick used above can be adapted by multiplying not by 2 but by *r* to show that the sum of the series is

*S* = $latex\frac{a}{1 – r}$.

Specifically, for the haggling problem, *a* was $24 and *r* was $latex-\frac{1}{2}$. Plugging those numbers into the formula gives *S* = $latex\frac{24}{\frac{3}{2}}$, which equals $16, as before.

For the fly problem, we have to work a bit to find the leading term, *a*. It’s the distance traveled by the fly on the first leg of its back-and-forth journey, so to calculate it 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 traveled $latex\frac{3}{3+2}$ of the initial 20-mile separation, which tells us *a* = $latex\frac{3}{5}$ × 20 = 12 miles. Similar reasoning reveals that the legs shrink by a ratio of *r*= $latex \frac{1}{5}$ each time the fly turns around. Von Neumann saw all of this instantly and, using the $latex\frac{a}{1 – r}$ formula above, he found the total distance traveled by the fly:

*S* = $latex\frac{12}{1-\frac{1}{5}}$ = $latex\frac{12}{\frac{4}{5}}$ = $latex\frac{60}{4}$ = 15 miles.

Now back to the larger point: How do series like this serve to connect the various parts of math? To see this, we need to enlarge our point of view about formulas like

1 + *r* + *r*^{2} + *r*^{3} + … = $latex\frac{1}{1-r}$,

which is the same formula as before with *a* equal to 1. Instead of thinking of *r* as a specific number like $latex\frac{1}{5}$ or $latex-\frac{1}{2}$, think of *r* as a variable. Then the equation says something amazing; it expresses a kind of mathematical alchemy, as if lead could be turned into gold. It asserts that a given function of *r* (here, 1 divided by 1 − *r*) can be turned into something much simpler, 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 come up virtually everywhere in science and engineering. The pioneers of calculus discovered that all the functions they were familiar with — sines and cosines, logarithms and exponentials — could be converted into the universal currency of “power series,” a kind of beefed-up version of a geometric series where the coefficients may now also change.

And when they made these conversions, they noticed startling coincidences. Here, for example, are the power series for the cosine, sine and exponential functions (don’t worry about where they came from; just look at their appearance):

$latex\cos x$ = 1 – $latex\frac{x^{2}}{2 !}$ + $latex\frac{x^{4}}{4 !}$ – $latex\frac{x^{6}}{6 !}$ + …

$latex\sin x$ = $latex x$ – $latex\frac{x^{3}}{3 !}$ + $latex\frac{x^{5}}{5 !}$ – $latex\frac{x^{7}}{7 !}$ + …

$latexe^x$ = 1 + $latex x$ + $latex\frac{x^{2}}{2 !}$ + $latex\frac{x^{3}}{3 !}$ + $latex\frac{x^{4}}{4 !}$ + …

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

That coincidence, and that kind of wishful thinking, led Leonhard Euler to the discovery of one of the most marvelous and far-reaching formulas in the history of mathematics:

$latexe^{ix}$ = $latex\cos x$ + *i* $latex\sin x$,

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

Euler’s formula expresses an outrageous connection. It asserts 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 the number *e* to an imaginary power (whatever that means). Euler’s formula, spawned directly by infinite series, is now indispensable in electrical engineering, quantum mechanics and all technical disciplines concerned with waves and cycles.

Having come this far, we can take one last step, which brings us to the equation often described as the most beautiful in all of mathematics, for the special case of Euler’s formula where *x* = π:

*e*^{iπ} + 1 = 0.

It connects a handful of the most celebrated numbers in mathematics: 0, 1, π, *i* and *e*. Each symbolizes an entire branch of math, and in that way the equation can be seen as a glorious confluence, a testament to the unity of math.

Zero represents nothingness, the void, and yet it is not the absence of number — it is the number that makes our whole system of writing numbers possible. Then there’s 1, the unit, the beginning, the bedrock of counting and numbers and, by extension, all of elementary school math. Next comes π, the symbol of circles and perfection, yet with a mysterious dark side, hinting at infinity in the cryptic pattern of its digits, never-ending, inscrutable. There’s *i*, the imaginary number, an icon of algebra, embodying the leaps of creative imagination that allowed number to break the shackles of mere magnitude. And finally *e*, the mascot of calculus, a symbol of motion and change.

When I was a boy, my dad told me that math is like a tower. One thing builds on the next. Addition builds on numbers. Subtraction builds on addition. And on it goes, ascending through algebra, geometry, trigonometry and calculus, all the way up to “higher math” — an appropriate name for a soaring edifice.

But once I learned about infinite series, I could no longer see math as a tower. Nor is it a tree, as another metaphor would have it. Its different parts are not branches that split off and go their separate ways. No — math is a web. All its parts connect to and support each other. No part of math is split off from the rest. It’s a network, a bit like a nervous system — or, better yet, a brain.