Gottfried Wilhelm Leibniz

Calculus, Infinite Series, and the Bernoullis

HON 201

Biography

Gottfried Wilhelm Leibniz (1646–1716)

Philosopher, diplomat, and co-inventor of calculus — and the infinite series that grew out of it.

A Universal Genius

• Leibniz worked across mathematics, philosophy, science, diplomacy, theology, ethics, politics, law, history, philology, and linguistics — and is sometimes credited as a forerunner of computer science.
• He wrote in Latin, French, and German, and also read English, Italian, and Dutch.
• He was a rationalist: he believed reasoning from clear, self-evident principles could reveal truths about the world.
• He hoped to build an “alphabet of human thought” — a symbolic system that could reduce complex reasoning to calculation.

Early Life

• His father was a professor with a large personal library, which Leibniz read through as a child.
• He entered university at 15, earned a doctorate at 20, and also held a law degree.
• In Paris, he met the Dutch scientist Christiaan Huygens — who had studied the cycloid and Saturn’s rings — and within four years became a mathematical star in his own right.
• Huygens received a pension from Louis XIV, who reigned as King of France from age 4 to 72 and had more than 19 children by seven different mothers.
• Huygens then set Leibniz a problem: find the sum of the reciprocals of the triangular numbers.

Louis XIV, King of France (Hyacinthe Rigaud, 1701).

Huygens is the connective thread to the earlier lecture on him: here Leibniz encounters Huygens directly, working in Paris under the patronage network of Louis XIV’s court.

The Mathematics

Leibniz’s Mathematics

• First to explicitly use the modern concept of a function, and to introduce terms like tangent, chord, abscissa, and ordinate.
• A pioneer of actuarial science — he calculated life annuities and the liquidation of debt.
• Introduced Gaussian elimination and the theory of determinants, anticipating Cramer’s rule for solving systems of equations with determinants.
• Co-invented calculus, alongside Newton, working independently.
• Found the series: \[\tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots\]
• Setting \(x = 1\) in his arctangent series gives a simple, famous formula for \(\pi\): \[\tan^{-1}(1) = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = \frac{\pi}{4}\]

Leibniz’s Calculating Machine

The Stepped Reckoner (Staffelwalze), Leibniz’s mechanical calculator, designed to perform addition, subtraction, multiplication, and division automatically — part of why he’s sometimes called a forerunner of computer science (see “A Universal Genius”).

An Infinite Sum

Sum of Reciprocals of Triangular Numbers

The triangular numbers count dots arranged in a triangle: \(1, 3, 6, 10, 15, \dots\) Huygens asked Leibniz to find:

\[1 + \frac{1}{3} + \frac{1}{6} + \frac{1}{10} + \frac{1}{15} + \dots = ?\]

• Leibniz’s trick: double the sum. Each term \(\frac{2}{n(n+1)}\) splits into a difference of two fractions, \(\frac{1}{n} - \frac{1}{n+1}\). Stacking these differences, almost everything cancels — a telescoping sum.
• What survives is just the first term, 1. Since that equals half the original sum, the full sum is 2.

\[1 + \frac{1}{3} + \frac{1}{6} + \frac{1}{10} + \dots = 2\]

Proving It Rigorously

The telescoping argument is persuasive, but Leibniz also gave a version that holds up to modern standards, using a finite partial sum and a limit.

1. Write \(S\) for the sum of \(\frac{1}{n}\) from 1 to \(n\), plus an extra \(\frac{1}{n+1}\) term, then subtract the same sum shifted by one.
2. Almost every term cancels, leaving: \(1 - \frac{1}{n+1}\) equals the sum of \(\frac{1}{k(k+1)}\) from \(k = 1\) to \(n\).
3. Taking the limit as \(n \to \infty\), the \(\frac{1}{n+1}\) term vanishes, leaving exactly 1.
4. Doubling (since the original sum is twice this one) gives 2 — confirming the earlier result, rigorously.

A note on caution: not every “obvious” infinite sum behaves this well — see the next slide.

Harmonic Series Diverges

Jacob Bernoulli proved that the harmonic series — the sum of every reciprocal, \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots\) — grows without bound, unlike Leibniz’s series.

The proof starts from half of Leibniz’s series and regroups it in a way that relates directly to the harmonic series.

Growth of the Harmonic Series

The harmonic series diverges — but very slowly.

It takes an enormous number of terms just to pass modest totals like 20 or 30. The sum is unbounded, but only just — it grows roughly as slowly as the logarithm of the number of terms.

The Bernoullis

Jacob Bernoulli (1654–1705)

• The elder of the two Bernoulli brothers, by 13 years.
• Defined \(e\) as the limit of \(\left(1 + \frac{1}{x}\right)^x\) as \(x \to \infty\).
• Made foundational contributions to probability, including the law of large numbers.
• Studied the lemniscate — the curve traced by inverting a hyperbola — now named after him.

Johann Bernoulli (1667–1748)

• Jacob’s younger brother, and also a major contributor to calculus and infinite series.
• Was hired to privately tutor the Marquis de l’Hôpital — and effectively wrote the first calculus textbook, which included l’Hôpital’s rule.
• The two brothers were fiercely competitive. After Jacob’s death, Johann grew jealous of Jacob’s son Daniel — and at one point falsely re-dated his own work on hydrodynamics to appear to predate his son’s.

Summary

• Leibniz developed calculus independently of Newton, along with much of the notation still used today (including the integral sign and \(d/dx\)).
• His work on infinite series — like the sum of reciprocals of triangular numbers — helped establish techniques (telescoping, partial sums, limits) still used to evaluate series rigorously.
• Jacob Bernoulli’s proof that the harmonic series diverges showed that not every well-behaved-looking series has a finite sum.
• The Bernoulli family, over several generations, produced some of the era’s most important mathematicians — not without rivalry along the way.