Let me be very clear, as clear as the vials of tears that I keep on my desk: This story is a long and sad one. It converges to no happy ending, and perhaps does not converge at all, although as you read, you will find your own joy and sanity both converging swiftly to zero.
If you were to abandon this text and go read about something pleasant, like butterscotch pudding or statistical sampling, I would applaud your good judgment, and humbly beseech you to statistically sample a pudding on my behalf.
As for me, I am compelled to tell this tale to its sour end, because I am an analyst—a word which here means “someone who fusses over agonizing details, bringing grief to many and enjoyment to none.”
But if you insist on reading further, then you ought to meet the three poor students at the heart of this tale:
Though all enrolled in the same course covering mathematical series, each came for a personal reason. Violet was drawn by the practical applications of series; Klaus, by their central role in the birth of modern mathematical thought; and Sunny, because this seemed the next logical step for her education, where “logical” means “expected by her parents and the HR departments at large corporations.”
Each came with good mathematical preparation and fresh enthusiasm. But their hearts all fell when they met Professor Olaf.
“We shall begin with arithmetic series,” said Olaf, not bothering to say so much as “hello” or “welcome to class” or “here are some reasons why learning about series is a good use of time that might otherwise be spent playing guitar or preparing lasagna or watching films about elephants.”
I am sorry to say that this is how teachers often are.
“Now,” said Olaf, “it is obvious that the sum of a finite arithmetic series is half the sum of the first and final terms, multiplied by the number of terms.”
“But why?” said Violet. “Can you offer some intuition?”
“Silence, undergraduate!” cried Olaf. “It is true for reasons any fool can see.”
“We can’t,” protested Klaus, “and we’re not fools.”
“Well,” said Olaf with malice, “I’m afraid you’ll need to find a fool to explain it to you. I cannot be bothered to hold the hands of dimwitted undergraduates orphans.”
Sunny rolled her eyes.
“Now,” said Olaf, “on to infinite geometric series.”
The word “geometric” here means “having nothing to do with geometry.” As you might guess, this topic offered no great clarity.
On it went. By the time the class ended, Olaf had filled the board with inscrutable notes about series of all kinds: arithmetic and geometric, convergent and divergent, harmonic and amelodic, alternating and direct, Taylor and Swift, McLaurin and McGuffin.
Then, for homework, he assigned a delta-epsilon proof, a phrase which here means “symbolic manipulations that persuade no one of anything.”
Now, if you are anything like me, you would have met this onslaught by filling new vials of fresh tears and filing a hasty application to change majors, perhaps to English or Art or simply Cowering Under the Bed Studies. But the three students, far braver than I, instead formed a study group.
Violet, pragmatic and efficient in the manner of all good engineers, found a trick for adding up certain long strings of numbers.
“Just imagine that there are two copies of the string,” she explained to the others. “And write the second one in reverse order, just below the first.”
“Now, we have a string of pairs, and each pair adds to the same amount.”
“This makes it easy to find the total simply by multiplying.”
Klaus and Sunny were very impressed, although Violet pointed out that this only works when the original string of numbers share a common difference. “Otherwise,” she explained, “each pair will add to a different total, and the whole method will unravel.”
(This trick, of course, had been known to many others, including the great Carl Gauss, and the far less great Professor Olaf. Indeed, beautiful ideas sometimes fall into the clutches of ugly minds, and do not always manage to brighten their surroundings.)
Meanwhile, Klaus—attracted by big ideas and conceptual shifts, as all great historians of mathematics are—found an infinitely long string of numbers whose total was, miraculously, not infinite.
“Look at this,” Klaus said, “and imagine the numbers go on forever, growing smaller and smaller.”
“Now, it might seem that infinite numbers should add up to infinity. But they don’t! Every new number brings us half of the remaining distance to 1. And so, no matter how far you go, the total can never exceed 1.”
“Instead, going further brings you closer and closer to 1, until you are less than a hair’s breadth. In some sense, the ‘final’ sum must be 1 itself, although this will happen only at the end of eternity, and that is well after our curfew.”
(In his exposition, Klaus joined many centuries of mathematicians who had puzzled over precisely this paradox—a word which here means “a mathematical oddity that prompts you to think two contradictory thoughts at once.”)
Finally, Sunny grabbed a piece of paper and wrote down the following strange observation:
“Really?” said Klaus.
“Why is that true?” said Violet.
Sunny shrugged and said, “Taylor,” which her companions understood to mean, “Proving this striking claim would require mathematical machinery beyond our command, but it certainly whets my appetite for the further study of series.”
“Wow,” said Violet as the study session ended, “we’ve learned a lot.”
“Yes,” said Klaus, “but we still haven’t completed the homework.”
Violet sighed. “I don’t understand deltas and epsilons at all.”
“Greek,” said Sunny. She did not mean “delta and epsilon are letters in the Greek alphabet,” although they are. Rather, she meant, “Don’t despair! The highly technical 19th-century framework of deltas and epsilons would have been alien to the inventors of calculus, and entirely baffling to the great Greek mathematicians of antiquity.”
“You’re right, Sunny,” Klaus said. “Even Cauchy, the scholar credited with our modern understanding of convergence, didn’t develop the language of deltas and epsilons. It’s no knock against us to stumble over these subtle, taxing ideas.”
“But what about Olaf?” Violet said. “And our homework?”
The three students put their heads together—a phrase which here means “exchanged ideas, without actually bringing their foreheads into contact”—and came up with a plan.
When they arrived at the next lecture, Olaf collected their homework. “What is this?” he said, gazing at the pages full of strange symbols.
“It’s an omega proof,” Violet said.
“And mine is a lambda proof,” Klaus added.
“Alpha,” Sunny said, describing her proof.
“You malodorous undergraduates!” roared Olaf. “I don’t know any of those proof styles! I only know about delta-epsilon proofs, because that’s what’s in the textbooks.”
“Greek letters are only symbols,” Violet said. “It doesn’t matter which ones you use.”
“Is your thinking so brittle that you can’t handle a mere change of notation?” Klaus asked.
“Alpha,” Sunny said, commenting on Olaf’s personality type.
“Omega proofs? Lambda proofs? Alpha proofs?!” Olaf cried. “I’ll tell you what I think of your precious Greek letters, you verminous undergraduates.”
And on a fourth piece of paper, he wrote a giant letter F.