Case Study 1 — How a Banking Question Discovered the Number $e$

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Case Study 1 — How a Banking Question Discovered the Number $e$

Field: Economics and finance (history of mathematics) Calculus used: Sequence limits (§20.3–20.4), the Monotone Convergence Theorem (§20.6), and the standard limit $(1 + x/n)^n \to e^x$ (§20.5, §20.8)

In 1683 the Swiss mathematician Jacob Bernoulli sat down with a deceptively simple question about money, and walked away having stumbled onto one of the two most important constants in all of mathematics. He did not set out to discover a fundamental number of the universe. He set out to figure out how much interest a bank should pay. That a question about a savings account should hand us the same constant that governs radioactive decay, population growth, and the bell curve of statistics is one of the quiet miracles of calculus — and it is, at its heart, a question about whether a particular sequence converges.

The Banker's Question

Imagine you deposit one dollar at an annual interest rate of 100%. (No real bank pays this, but the round number makes the arithmetic transparent; everything scales to realistic rates at the end.) Compounded once a year, at the end of the year you have

$$1 \times (1 + 1) = 2 \text{ dollars}.$$

Now a clever customer objects: "Why wait a whole year? Pay me half the interest after six months, and let that interest earn interest for the second half." Compounding twice means applying 50% growth twice:

$$\left(1 + \tfrac12\right)^2 = (1.5)^2 = 2.25 \text{ dollars}.$$

The customer earned an extra 25 cents simply by collecting interest sooner and letting it compound. Greedy now, the customer demands quarterly compounding — 25% applied four times:

$$\left(1 + \tfrac14\right)^4 = (1.25)^4 = 2.44140625 \text{ dollars}.$$

Monthly compounding (12 times, at $100\%/12$ each):

$$\left(1 + \tfrac{1}{12}\right)^{12} \approx 2.6130 \text{ dollars}.$$

The pattern is clear: the more often you compound, the more you earn. So Bernoulli asked the natural — and dangerous — question. What if you compounded continuously? Every instant, infinitely often, an infinitesimal sliver of interest added and immediately set to work earning more. If compounding more often always pays more, does compounding infinitely often pay infinitely much?

The Sequence at the Center of It

Strip away the dollars and you are looking at a sequence. Let $n$ be the number of compounding periods per year. After one year, one dollar at 100% becomes

$$a_n = \left(1 + \frac{1}{n}\right)^{\!n}.$$

This is precisely the object §20.3 taught us to interrogate: an infinite list of numbers, and we want to know whether it settles down. Let us watch it climb as $n$ grows.

# One year, rate 100%, compounded n times: a_n = (1 + 1/n)^n
for n in [1, 2, 4, 12, 365, 8760, 525600]:
    print(f"n = {n:>7}:  (1 + 1/n)^n = {(1 + 1/n)**n:.8f}")

# n =       1:  (1 + 1/n)^n = 2.00000000   (annually)
# n =       2:  (1 + 1/n)^n = 2.25000000   (semiannually)
# n =       4:  (1 + 1/n)^n = 2.44140625   (quarterly)
# n =      12:  (1 + 1/n)^n = 2.61303529   (monthly)
# n =     365:  (1 + 1/n)^n = 2.71456748   (daily)
# n =    8760:  (1 + 1/n)^n = 2.71812669   (hourly)
# n =  525600:  (1 + 1/n)^n = 2.71827924   (every minute)

Look carefully at the right-hand column. The balance rises every time we compound more often — but the increments are shrinking. Going from annual to semiannual buys you 25 cents. Going from daily to hourly buys you about a third of a cent. Going from "every minute" to "every second" would buy you almost nothing. The sequence is increasing, but it is being throttled. It is converging.

Why the Limit Is Finite — and Why That Needed Proving

Here is where the customer's greedy intuition fails and §20.6 rescues us. The naive expectation — multiply $n$ factors each bigger than $1$, get something enormous — ignores that the factors themselves shrink toward $1$ as $n$ grows. The two effects fight, and it is not obvious which wins. This is exactly the kind of indeterminate tug-of-war ($1^\infty$) that demands the careful machinery of limits rather than hand-waving.

The Monotone Convergence Theorem settles it. One can show two facts about $a_n = (1 + 1/n)^n$. First, the sequence is increasing — each extra compounding event adds a little "interest on interest," nudging the balance up. Second, it is bounded above — a binomial expansion of $(1+1/n)^n$ shows every term is dominated, factor by factor, by the corresponding term of $1 + 1 + \tfrac{1}{2!} + \tfrac{1}{3!} + \cdots$, a sum Bernoulli could bound below $3$. Increasing and bounded above means convergent, by the MCT of §20.6 — and crucially, the theorem guarantees a finite limit exists without telling us its value. That guaranteed limit is named:

$$\lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^{\!n} = e \approx 2.718281828\ldots$$

This is one of the standard definitions of Euler's number $e$. Bernoulli computed enough terms to see the convergence and bracket the answer between $2$ and $3$; he did not have a closed form. The symbol "$e$" and the deep theory came later, from Leonhard Euler in the 1730s, who showed the same number arises as the infinite sum $e = \sum_{k=0}^\infty 1/k!$ (the Taylor series we reach in Chapter 23) and proved the general law

$$\lim_{n\to\infty}\left(1 + \frac{r}{n}\right)^{\!n} = e^{r}.$$

That generalization is the bridge from a curiosity to a workhorse. It says: continuous compounding at annual rate $r$ turns one dollar into $e^r$ dollars after one year, and into $e^{rt}$ dollars after $t$ years.

From Thought Experiment to Your Bank Statement

Real banks do not pay 100%, but the same sequence governs the fine print on every account. Suppose an account advertises a 6% annual rate. Compounded monthly, one dollar grows by a factor of

$$\left(1 + \frac{0.06}{12}\right)^{12} = (1.005)^{12} \approx 1.061678,$$

an effective annual yield of about 6.168% — slightly more than the stated 6%, because the monthly interest compounds. Push compounding to its continuous limit and the factor becomes

$$e^{0.06} \approx 1.061837,$$

a yield of about 6.184%. The gap between monthly and continuous compounding here is roughly two thousandths of a percent — economically negligible, which is exactly why the sequence converging matters in practice. Continuous compounding is not some exotic banking product; it is the clean mathematical idealization that monthly or daily compounding already nearly achieves. Financial models reach for $e^{rt}$ not because banks literally compound every instant, but because the limit is both simple and accurate.

That single limit, born from Bernoulli's dollar, now sits at the foundation of quantitative finance. Present-value and bond-pricing formulas discount future cash flows by $e^{-rt}$. The Black–Scholes option-pricing model — the equation that built modern derivatives markets — uses $e^{rt}$ for its risk-free growth factor. Step outside finance and the same $e^{rt}$ describes exponential population growth, the charging of a capacitor, and (with a negative exponent) radioactive decay and drug elimination. All of it traces back to one convergent sequence.

Why It Had to Be a Limit

It is worth pausing on what calculus contributed here. A purely arithmetic mind can compute $(1 + 1/n)^n$ for any specific $n$ — Bernoulli did, by hand. But no finite computation can answer "what happens at infinity." That question only has meaning, and only has an answer, because the limit concept of §20.3 makes "compound infinitely often" precise as "the limit of the sequence as $n \to \infty$," and the Monotone Convergence Theorem of §20.6 guarantees that limit is a genuine real number rather than a runaway to infinity. The number $e$ is not something humans invented and then noticed in interest rates. It is the answer the sequence was always going to give, waiting to be discovered the moment someone asked the question correctly.

Discussion Questions

Why doesn't the product diverge? Each factor $(1 + 1/n)$ exceeds $1$, and you multiply $n$ of them. Explain, using the shrinking-increments column in the code output, why the result stays finite. Which theorem of §20.6 makes this rigorous?
Increasing and bounded. State precisely what the Monotone Convergence Theorem requires, and identify the two facts about $(1+1/n)^n$ that supply those hypotheses. Why does the theorem let us conclude convergence without first knowing the value $e$?
From rate to yield. For a 4% nominal rate, compute the effective annual yield under daily compounding $(1 + 0.04/365)^{365}$ and under continuous compounding $e^{0.04}$. How large is the difference, and what does its smallness say about why $e^{rt}$ is a good modeling choice?
General growth. Using $\lim_{n\to\infty}(1 + r/n)^n = e^r$, find the continuous-compounding balance of \$1{,}000 invested at $r = 0.05$ for $t = 20$ years via $1000\,e^{rt}$. Compare to annual compounding $1000(1.05)^{20}$.
Discovered, not invented. The case study claims $e$ was discovered through the interest question rather than invented to model it. Do you agree? What would distinguish discovery from invention here?

A Short Annotated Reading

Maor, E. (1994). e: The Story of a Number. Princeton University Press. The definitive accessible history; Chapter 3 tells the Bernoulli compound-interest story in full.
Dunham, W. (1999). Euler: The Master of Us All. MAA. How Euler took Bernoulli's limit and built the modern theory of $e$, including the series $e = \sum 1/k!$.
Strogatz, S. (2019). Infinite Powers. Houghton Mifflin Harcourt. A chapter on $e$ that connects compound interest to the broader role of exponential functions in calculus.

A banker's question about a single dollar produced a number that runs through finance, biology, and physics alike. The bridge between the practical question and the universal constant is a single convergent sequence — and the Monotone Convergence Theorem that guarantees it lands on a real number.