The number e — limit and compound interest

What just happened?

You moved the slider to very frequent compounding. The capital after one year kept growing, but ever more slowly — the multiplier did not “escape” to infinity, it settled right next to one specific irrational number.

Compound interest and the limit of a sequence:

At an interest rate of 100% per year, compounding n times per year, the capital after one year is $(1+1/n)^n$ times the starting amount. For n=1 that is 2×, for n=12 already about 2.613×, and as n→∞:

— the sequence is increasing and bounded above, so it has a finite limit. That limit is precisely the number $e$.

Where did the number e come from?

Jacob Bernoulli stumbled upon this constant at the end of the 17th century, studying exactly this problem: how much do we earn by compounding interest ever more often? He noticed the result converges to some limiting value, but could not pin it down exactly. Only Leonhard Euler in the 18th century — in his work „Introductio in analysin infinitorum" (1748) — gave the constant the symbol “e”, computed it to high precision (18 decimal places), and showed its fundamental importance in mathematics.

Why does it not grow to infinity?

More frequent compounding does increase the capital — but the gain from each additional compounding gets smaller and smaller, because we split the same 100% across more and more, ever shorter periods. The sum of these diminishing gains is bounded, so the sequence $(1+1/n)^n$ grows yet converges to a constant — it does not explode.

Why this matters for your exams:

e is the base of the natural logarithm $\ln$ and one of the most important mathematical constants alongside π. It appears in differential calculus — the function $e^x$ is the only function equal to its own derivative: $(e^x)' = e^x$ — as well as in statistics (the normal distribution), physics, and many other fields.

Remember: $(1+1/n)^n \to e \approx 2.71828$ as n→∞ — this limit is always exactly the same, regardless of how fast n grows to infinity.