Volume 6, Entry 14: Programming

Infinite series, counting to a million, and me

I was seventeen when Mr. Richmond told my class that the harmonic series diverges.

Truthfully, he probably didn’t use those words. There’s a good chance his lesson that day was fully exploratory, that he didn’t name the series even once and didn’t use the technical term diverges because the vocabulary wasn’t the point. But the idea was there. We were adding it up and talking about infinity.

For the uninitiated (or extremely rusty), the harmonic series adds up the terms of a straightforward sequence of numbers: the reciprocals of the positive integers. That is, we express the harmonic series like this:

When you sum up numbers from an ordered sequence like this, the result is called a series, and the harmonic series is among the most famous in mathematics. The harmonic series’ elegant simplicity is only outdone by its significance.

But we’re in Honors Pre-Calculus during the 2002-2003 school year right now, so that significance isn’t the point. The actual point: this series never stops growing. In other words, if you could keep adding together the infinity terms comprising it, the sum itself would be infinite.

If this doesn’t surprise you, that’s fine; we’re adding up an infinite collection of positive numbers, so their sum growing without bound comes fully loaded intuitionally. Adding more always means more, right?

Well, yes and no. Yes, adding more does mean more, but that doesn’t always mean you keep growing to infinity. Sometimes, if you add up an infinite number of positive numbers, the sum will approach a finite value. For instance, this geometric¹ series

does not grow without bound. If you keep adding more and more terms in this pattern, you’ll find the sum approaches one instead. Again, not that the jargon matters here, but this series converges to one. If we could add every one of those positive terms, their sum would never exceed uno. It would just keep getting closer and closer and closer.

That Richmond started there—with the geometric series with those powers of two—ignited an intuition in my teenage brain. It said, counterintuitively, An infinite set of positive numbers can add up to something finite if its terms get really, really small. I mean, that’s what’s happening with the 1/2 series: those terms get so small as we proceed that it curbs the growth. Today, I’d describe that sum of one as asymptotic, but I hadn’t cemented that word in my lexicon then. I developed understanding as Richmond spoke; my intuition surged that day while he narrated at the whiteboard.

At least until the harmonic series entered the ring. Because after talking about that geometric series, he baited me into a common misconception: if

then

must equal some number as well, right?

Except, Richmond noted, it doesn’t. The first series sums to one, but the second series—the harmonic series—sums to infinity. It never stops growing.

I remember precisely how my brain responded when he said this. That might seem unlikely, but it’s easy to remember because, amid all this talk of infinity, only one word entered my brain.

Bullshit, I thought. Bullshit.

He insisted the harmonic series would grow forever, and I didn’t believe him. He would go on to demonstrate his assertion on a small scale, claiming that the sum could exceed any positive integer while using a now-ancient television connecter for his calculator, but my skepticism remained. The idea violated my intuition. Those terms get so small! How could adding increasingly tiny numbers so clearly approach a value in one situation and break a supercomputer in another?

As I tended to do when I didn’t believe something in a math class, I used technology to run an experiment. My first programming experience was HTML during elementary school, but Richmond taught us the basics of programming the previous year. Because I had a different calculator from everyone else—a TI-86, the black sheep of the company’s flagship product line—and no YouTube to guide me, I’d had to fiddle around to keep up, which stoked my curiosity and led to exploration and a resultant surge in understanding. I leaped from the barebones Quadratic Formula program we made in class to a complex triangle-solving application of my design. Forget Harry, Hagrid—I was a wizard with that device.

So I put that wizardry on display in Richmond’s class that day. Determined to show that the sum eventually settled in at some value, I devised a looping program that computed and displayed partial sums of the harmonic series in rapid succession. These values would appear one after another, letting me monitor the series for potential convergence.

With about sixty minutes left in class, I initiated my program and watched an endless column of irrationals cascade across my screen. I flipped it over on my desk, confident that I'd see the values flatlining when the bell rang to conclude the day.

One hour later, a familiar tone sounded to release us. I picked up my calculator, ready to race over to Richmond and show him what I’d found. But I never did. The reason was simple: despite 30,000+ calculations, the sum was still growing. Unequivocally growing.

No matter what I believed, the harmonic series showed no signs of stopping.

*****

It’s twenty-two years later, and I now possess proof that the harmonic series diverges.

Technically, I got my first proof one year after Richmond’s introduction. In Calculus, Mr. Friedrich taught us the Integral Test, a diagnostic for specific series. During that lesson, he used the technique to prove that the harmonic series diverges to infinity. Although it would be a decade before I truly understood (and fully trusted) the Integral Test, that was helpful: no matter what my intuition said, I could lean on that straightforward, if cumbersome, method to reaffirm Richmond’s notion.

In 2018, a student convinced me. DeVon, himself curious, uncovered an argument online that shattered my intellectual skepticism. I’ll spare you the details here², but it involves comparing something clearly infinite to the harmonic series. The moment he finished explaining it to me, a fog lifted. His proof wasn’t presented with intense rigor—he described it verbally, with his trademark earnest energy—but the rationale made sense. He did what my calculator, teachers, and textbooks never could: convince me.

It’s such a wild thing to undergo that transformation. There are light years between unsure and ironclad, yet it sometimes takes only a few words to flip from one to the other. That brief conversation with DeVon transported me across space and time. His explanation clicked so completely that I narrated his approach to both classes the next day. Although the fact feels unintuitive, the harmonic series’ infinite sum became unimpeachable after DeVon. I had no more doubts. I believed him.

Unfortunately, belief of that caliber doesn’t come naturally to me. Whereas I can be convinced of a mathematical result by a sound argument from a reliable resource, I find myself incapable of the same in other realms.

My typical word for this is slippery. My belief in things is slippery, I’ll say. What is “belief in things”, you might ask?

Everything. Everything.

My confidence in friendships is slippery. Spending time with people equips me with evidence that I don’t matter. Listening carefully and actively to them, I clarify and summarize regularly, intent on making them aware of my attentiveness. But when the talking stick travels my way, I can tell people don’t care. Their phones appear after one sentence. Vague, out-of-rhythm Mhmms respond to even my most salient points. Half the time, I end outings feeling like an anthropomorphic acoustic pad.

My confidence in my instruction is slippery. No matter what element I add or approach I take, I look out at the classroom and find a wealth of disinterest and disrespect. I understand the YouTube videos and NCAA tournament games are more interesting than the math I’m delineating; it’s not their desire to do other things that bothers me. It’s that I haven’t secured their respect, that the wealth of resources, the rec letters, and the support I offer aren’t enough. Those things aren’t meant to be transactional, I know, but I can’t help but feel my investment doesn’t mean anything, so my teaching doesn’t either. I feel stupid and small. I’m a children’s clown hired for a sweet sixteen.

My confidence in my writing is slippery. Between Grammarly’s 196 weekly suggestions and a platform that hurls unsolicited advice at me each time I open the app, enough self-assurance leaks out that it begs to be a related rate problem. Abandoning a complete draft used to be an isolated incident; these days, it’s a rare week when I don’t scrap one or two, my mind panicking over increasingly tiny concerns. And it spreads from here: Every sentence of my manuscript reeks like diced ham from fourteen weeks ago, and every new idea I conjure feels tired or tepid (or both).

My confidence in myself is slippery, too. Especially so. Nothing I say sounds meaningful. My gut feels invisibly distended below its surface, my charisma plays on tape delay, I third-guess every instinct I have and decision I make like I’m under cross-examination. I’m empty of emotion when I need it and overflowing when I don’t. Quiet desperation forms the quaking bedrock under every moment and interest in my life. I used to get space from my natural nihilism, the rougher bouts leading to long periods with it locked away, but the spiral’s tighter now. As soon as I exit a crater, I lose my balance and start to fall back in.

And yet, despite all that slippage, I can counter every point. I’ve had rewarding conversations with friends recently, the long walks, impromptu phone calls, heartening comments, and crisp french fries providing peace. Ditto for school: I have a class that is so active and friendly that it feels like 2019. I look around the room during lectures and find thoughtful eyes; they whisper insight at one another and ask so many questions that the lessons feel fresh. My writing still rewards me on its own, even when it devolves into frustration-fueled resets, and the weekly notifications of a comment from Michael or a like from Abby straighten my literary spine. And internally? Well, internally, I’m a mess, but I’m still recording Daily Positivity. While there have been flippant nothing-burger transcriptions in 2025, most capsules exceed 120 words. I know there’s good out there, and usually, in the early morning, I can even trace it.

But these things, while each wonderful, amount to the Integral Test. I comprehend their rigor in a vacuum: I concede that they prove the harmonic series diverges. But they fail to sway me. They are apparent antidotes to the poison my self-worth secretes, but they never stay in my system. I’m never convinced; I’m constantly calling bullshit on my positive outlook.

That’s a problem because I can feel a change taking hold inside me. I’m excusing myself from interactions I usually look forward to, taking fewer outdoor walks than ever, and putting off fulfilling calls. Last quarter, I seriously entertained adopting a wholly digital lesson delivery, a move that would transform personable instruction into rote grading in a silent room. Several times each week, I retreat to my digital journal, writing for an audience of one because words and ideas flow there unencumbered by my fear over how others will receive them. As for the fourth front, I’m overwhelmed but still volunteering for more, one of the tell-tale signs I’m down on myself. When I feel repulsive and weak, I overextend to justify my existence.

This is me in Pre-Cal again. I’m reaching for my TI-86 and hastily coding programs to counter a trusted source instead of listening to his thoughtful overview. Could Richmond have presented an argument similar to DeVon’s after making his claim about the harmonic series? I want to say no, but it took me some time to code the thing from scratch and test out the Lbl and Goto commands that day. Maybe he addressed the tricky part perfectly, and I missed it while under the influence of incorrect intuition? It wouldn’t be the first time—I’m a highly trained tree-seeing forest-misser.

Still, while I worry about many things, my defining disbelief isn’t one of them. Trusted people pointing out red roses that look blue to me feels normal; obsessing over three tiny moments in an otherwise successful day marks my default state. I’m used to doubt, and I’m used to the self-inflicted wounds doubt leaves behind. I draw strength from my resilience in the face of it all; pulling myself out of quicksand leaves me proud, even when I made the perilous pit with my own two hands.

I’ve lived with this uncharitable mindset toward myself for so long that I’m used to it. Who cares that you find my syntax strange and my menus messy? Navigating my OS is easy when it’s the only one I’ve ever known. I’m a TI-86, baby—I present complex numbers as ordered pairs and omit the stat package entirely because reasons. But this whiz can do wild things with an inferior product. I’ve got too many calculations due to worry about the device I’m doing them on. Keep your 89’s symbolic logic and your 84+ CE’s color display. I’ve got lowercase letters and sophisticated vectors all to myself. Why stress over a device I know backward and forward?

There’s nothing to worry about here. So what if overcoming my crippled self-worth makes every task feel like two? If a self-sabotaging cynic struggles but gets everything done anyway, what’s the big deal? Inefficiency isn’t a crime! Suboptimal devices are allowed on the AP Calculus exam!

I’ve always had this disbelief, but I’ve likewise always found a way to scratch out a little more and keep going despite myself. Perhaps I’m like the harmonic series in that way: no matter how small I feel, there’s always enough to eke out just a little more. Until I find my DeVon ex machina that magically incepts the confidence I lack, I’ll be computing sums on a device you’d never buy.

Think what you want about the way I do things, but I’m durable and reliable and capable of some crazy cool shit. Once again, I’m like the TI-86: I work in weird ways, but I get shit done.

Until I don’t, what’s there to worry about?

*****

Richmond had me back on my calculator a few weeks after the harmonic series episode.

I don’t remember what lesson it related to. If I had to guess, I’d point at Riemann Sums and their infinite number of rectangles as the culprit, but that would be a guess. Still, I remember his comment that sent my mind whirring.

“One million’s a huge number. It’d take a long time to count to a million.”

This claim incited the cash grab sequel to the harmonic series scenario. My brain immediately called BS on his innocuous claim, and I sought to disprove him before his next sentence could begin. Once again, I put my calculator to work.

As class ended, I put the finishing touches on a short program that resembled my harmonic series one. Truthfully, it was much simpler: all it did was count up. Line after line, it would cycle through simple code:

Input A
1->A
Lbl M
A+1->B
Disp B
B->A
Goto M

When the bell rang, I showed my calculator to Richmond.

“See, it won’t take that long,” I told him, pointing toward the integers flying up my screen. “Look at how fast it’s going!”

He chuckled but gave a generous shrug. “Maybe. You’ll have to see.”

Tucking the thing safely in my backpack, I let it run for the rest of the afternoon. Whatever I had going on that day—probably baseball or a shift at the card shop—held my attention, and I completely forgot about the program until the following day.

At 6:30 am, while eating a Pop Tart, I suddenly remembered, prompting me to swear under my breath. I expected to be ten million numbers deep by then—ten million at least. I wouldn’t be able to pinpoint precisely when it had crossed into seven digits as planned, but maybe I could estimate that interval from where it was when I looked.

Sliding my TI-86 out of my backpack, I unsheathed the thing and looked down at my screen, expecting to see eight digits. Squinting, I did a double-take.

It had just crossed 976,000.

After running endlessly for sixteen hours, my calculator still hadn’t reached one million.

To say this shocked me understates things. The sluggish progression floored me. I felt a touch of embarrassment even. Richmond hadn’t rigorously defined “long time”, but I felt safe anointing sixteen hours as “long”. My intuition the day before was dead wrong.

Eventually, of course, the TI-86 did cross 1,000,000. I let it keep going a little longer so Joey could see it on the ride to school, but I stopped the program before first period. That forced quit left the last numbers visible on the screen, so I could show Richmond.

I didn’t need my calculator until his class during sixth period, so it was only then that I withdrew it again. Walking up to his desk, I reflected on the entire situation. Although being so woefully wrong wounded my pride, I felt stronger for the experience. I understood scale in a way I never had before, the enormity of one million clearer than ever. I’d been clever in quickly designing a program, I’d been mature in conceding defeat, and I’d get to demonstrate and relate all of that to one of my most revered role models. My misplaced doubt and incorrect intuition had, ultimately, cost me nothing. I might’ve even learned some things.

So I arrived at Richmond’s desk and waited while he answered a classmate’s question. When he finally dispatched that student’s misconception, I leaned in with my calculator.

“When did it finish?” he asked, a knowing smile in accompaniment.

“Like sixteen hours later,” I said, pressing the on button as I did. “I couldn’t believe it. Here’s where I stopped it this morning.”

I handed him my TI-86. He squinted at the screen, pressed a button several times, and then raised an eyebrow.

“It won’t turn on,” he said, handing the graphing calculator back to me. And then he laughed. “Looks like all that counting killed your batteries.”

Returning to my desk in a whimpering daze, I kept jabbing the on button, hoping to see my screen come alive. But the tapping was to no avail. I had a dead device on my hands.

Right then, I learned another lesson, this one about computational cost: exerting energy to disprove things takes a toll. It seemed I had paid a price for my disbelief after all.

In 2003, that price was four AAAs and an Honors Pre-Cal lesson spent without a functioning calculator.

I wonder what my internal doubt will set me back in 2025.

(And beyond.)

---

I got stuck in the middle of this one for a little while. I decided to lean into a graphing calculator metaphor, a strategy I first saw in a(n excellent) piece of writing by a student. Unlike their essay, mine is the weakest part of this piece, but that’s okay. I enjoyed trying to make it work.

Although I talk about “math as metaphor” on occasion, this one truly arose out of the math: I taught this lesson on March 27th, and the memories from Richmond’s class seized my attention afterward. I doubt I’ll lean this heavily into something this technical again for some time, but since this stuff is my work, there’s something satisfying in this otherwise messy piece.

Amusingly, most of my writing this week was actually more related to the piece “Mutual Feeling” from two weeks ago: I’m about 80% of the way through an original Cross the Line for the new event my team and I are hosting for twelfth graders. It’s been a rewarding experience, and I think a part of that owes to reflecting on the activity.

1. Meaning you multiply by the same thing to generate each term from the previous

2. But not here!