Seven Proofs That the Harmonic Series Diverges

There's a particular flavor of mathematical cruelty embedded in the harmonic series: 1 + 1/2 + 1/3 + 1/4 + 1/5, on and on forever, each new term smaller than the last, shrinking toward zero—and yet the whole thing blows up to infinity. It looks like it should settle down. Every instinct you've developed about diminishing quantities says it should. It doesn't.

This is why the harmonic series is the first thing in a second-semester calculus course that genuinely surprises students. And it's also, apparently, why the standard classroom treatment bothers mathematician Michael Penn enough to frame a 31-minute video around it—titled, with a typo [sic] that reads more like a provocation, "you [sic] class covered this the worst way..." (YouTube). Penn's framing positions the most common textbook proof as the floor of understanding, not the ceiling. The video then works through seven distinct strategies to demonstrate the same result, each one revealing something different about why the series misbehaves.

That's the premise worth sitting with: one mathematical fact, seven proofs. Not because mathematicians are indecisive, but because each proof illuminates a different corner of the same truth.

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Penn opens with the grouping method, which dates to the medieval French philosopher Nicole Oresme—and yes, 1350 is the rough attribution. The move is elegant: you bracket the harmonic series into chunks of doubling size (1/3 and 1/4 together, then 1/5 through 1/8, and so on), then replace every term in each bracket with the smallest term in that bracket. This makes each bracket worth exactly 1/2, and since you can keep adding brackets forever, the series diverges. Penn shows this produces a lower bound of 1 + m/2, where m is the exponent of the largest power of two not exceeding n—and as n grows, so does m.

"This is the way that is generally presented in most calculus classes," Penn acknowledges, which is precisely why he doesn't stop there. The Oresme proof is clean and teachable. It's also, in his framing, a little too tidy—it tells you the series diverges without telling you much else about why or what structure underlies that divergence.

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The Cauchy condensation test arrives second. It's less a proof than a redirect: the test states that a decreasing series of non-negative terms converges if and only if a related "condensed" series converges. Apply it to 1/n and the condensed series becomes the sum of 1 for all k—which obviously diverges. Simple, almost anticlimactic, but it's the kind of tool that earns its place in real analysis courses precisely because it generalizes.

Third comes the integral test, and here Penn makes a visual argument that I find genuinely satisfying. He draws rectangles of width 1 and heights 1, 1/2, 1/3, ... over the curve y = 1/x, observes that each rectangle overestimates the area under the curve on its interval, and concludes that the sum of the rectangles (the harmonic series partial sums) must be at least as large as the integral of 1/x from 1 to n+1—which is ln(n+1), which goes to infinity. The geometry does the heavy lifting.

The fourth method takes a detour through the inequality e^x > x + 1 for all positive real x. Set x = 1/n, take logarithms, and you get 1/n > ln(n+1) - ln(n). Sum both sides from 1 to N and the right side telescopes to ln(N+1)—which again runs off to infinity, dragging the partial sums with it. This proof and the integral test "end up in the same place," as Penn notes, but they travel different roads to get there.

The fifth method uses a telescoping product. Penn constructs the product ∏(1 + 1/n) from n=1 to N, shows it telescopes to N+1, then loops back to the e^x > x + 1 inequality to establish that N+1 < e^(harmonic partial sum). Taking logarithms: ln(N+1) < 1 + 1/2 + ... + 1/N. Since ln(N+1) diverges, so does the partial sum. It's the same inequality doing double duty, which is the kind of mathematical recycling that feels almost satisfying.

The sixth method is my favorite of the bunch, and Penn calls it proof by contradiction. Suppose the harmonic series converges to some real number S. Split S into its odd-indexed terms (S_odd = 1 + 1/3 + 1/5 + ...) and even-indexed terms (S_even = 1/2 + 1/4 + 1/6 + ...). Factor 1/2 from S_even and you get the original series back—so S_even = S/2, which means S_odd = S/2 as well. But then Penn observes that every odd term dominates the corresponding even term (1 > 1/2, 1/3 > 1/4, and so on), so S_odd > S_even. That gives S/2 > S/2, which is a contradiction. The series cannot converge to any finite number.

"Which doesn't make any sense that S is bigger than S," Penn says, with the quiet satisfaction of someone who's just closed a trap.

The Prime Number Finale

The seventh proof is where things get genuinely strange and wonderful. Penn uses the fundamental theorem of arithmetic—the fact that every natural number factors uniquely into primes—to rewrite the harmonic series as an infinite product over all primes p:

∏_p (1 + 1/p + 1/p² + 1/p³ + ...)

Each factor in that product is a geometric series summing to 1/(1 - 1/p). So the harmonic series equals the product over all primes of 1/(1 - 1/p). Now: every one of those factors is greater than 1. A product of infinitely many terms, each greater than 1, diverges. Therefore, the harmonic series diverges.

What this proof is really doing is connecting the harmonic series to the distribution of prime numbers—the same connection that anchors the Riemann zeta function and, by extension, some of the deepest unsolved problems in mathematics. The proof that there are infinitely many primes goes back, according to Britannica, to Euclid around 300 BCE. Penn's final proof doesn't just confirm that the harmonic series diverges; it situates that divergence inside the infinite supply of primes. Run out of primes, and maybe the product closes. But you never do.

There's a pedagogical argument implicit in everything Penn does here, and it's worth naming: a student who only knows the Oresme grouping proof knows that the harmonic series diverges. A student who works through all seven knows how divergence is woven into the structure of the integers, the exponential function, geometric series, and the primes themselves. That's not the same knowledge, even if the conclusion is identical.

The question Penn leaves hanging—what's your favorite proof, and do you know one he didn't cover—is a genuine one. Seven routes to the same infinity, and mathematicians have been finding new ones for almost seven centuries. The series keeps diverging. The proofs keep multiplying.

— Nadia Marchetti, Unexplained Phenomena Correspondent, BuzzRAG