The number e — definition as limit of (1+1 - n)ⁿ, natural growth context

What $e$ is

WHAT each piece means:

$\frac{1}{n}$ = size of each interest chunk (if compounding $n$ times a year at 100%).
The power $n$ = how many times we compound.
Taking $n\to\infty$ = compounding continuously (all the time).

Deriving it from scratch (compound interest)

HOW growth builds up. Start with 1 unit, compound $n$ times at rate $x$ per year:

$\text{after 1 chunk} = 1\cdot\left(1+\tfrac{x}{n}\right)$ $\text{after 2 chunks} = \left(1+\tfrac{x}{n}\right)^2 \quad\text{(each chunk multiplies again)}$ $\text{after all } n \text{ chunks} = \left(1+\tfrac{x}{n}\right)^{n}$

Why this step? Because compounding is repeated multiplication, not addition — that's the whole point of "interest on interest."

Now let $n\to\infty$ to model continuous growth. The limit defines $e^x$ .

Derivation of the series (Why?): By the binomial theorem, $\left(1+\tfrac1n\right)^n=\sum_{k=0}^{n}\binom{n}{k}\frac{1}{n^k}=\sum_{k=0}^{n}\frac{n(n-1)\cdots(n-k+1)}{k!\,n^k}.$ The fraction $\frac{n(n-1)\cdots(n-k+1)}{n^k}=1\cdot(1-\tfrac1n)(1-\tfrac2n)\cdots\to 1$ as $n\to\infty$ . Why this step? Each factor $(1-\frac{j}{n})\to 1$ , so term $k$ tends to $\frac{1}{k!}$ . Summing gives $\sum \frac1{k!}$ . This series converges fast, giving us a way to actually compute $e$ .

Why does it converge instead of blowing up?

Figure — The number e — definition as limit of (1+1 - n)ⁿ, natural growth context

Worked examples

Worked example Example 1 — Watching the limit converge

Compute $\left(1+\frac1n\right)^n$ for a few $n$ .

$n$	$\left(1+\frac1n\right)^n$
1	$2.000$
2	$2.250$
10	$2.594$
100	$2.705$
1000	$2.717$

Why this step? Plugging concrete $n$ shows the sequence increasing but slowing toward $2.71828$ — the definition in action, not just symbols.

Common mistakes

Active recall

Recall Test yourself (open after attempting)

Define $e$ as a limit. → $e=\lim_{n\to\infty}(1+1/n)^n$ .
Why doesn't it diverge? → Correction terms $\frac1{k!}$ shrink super-fast; bounded above by 3.
What form is the limit? → $1^\infty$ indeterminate.
How do you get $e^x$ ? → $\lim_{n\to\infty}(1+x/n)^n$ .

Recall Feynman: explain to a 12-year-old

You put £1 in a magic bank that promises to double your money in a year. But you're greedy: instead of waiting the whole year, you ask for half the growth after 6 months, then let that grow too. You get a bit more than £2. Ask for it in monthly bits — a bit more again. Do it every second, every instant... you'd expect huge money, but it stops at about £2.72. That special stopping number is called $e$ . It's how fast things grow when they grow "as smoothly as possible."

Flashcards

What is the definition of

e

as a limit?

e=\lim_{n\to\infty}\left(1+\frac1n\right)^n\approx 2.71828

Why is

\left(1+\frac1n\right)^n

not just equal to 1?

It's a

1^\infty

indeterminate form — base

\to1

but exponent

\to\infty

; they balance at

e

Give the series expansion of

e

e=\sum_{k=0}^\infty \frac1{k!}=1+1+\frac1{2!}+\frac1{3!}+\cdots

Why does the series/limit converge?

Terms

\frac1{k!}

shrink faster than geometrically; sum bounded above by 3.

Formula for

e^x

as a limit?

e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n

Continuous compound interest formula and where

e

comes from?

A=Pe^{rt}

, the limit of

P(1+r/n)^{nt}

n\to\infty

e

rational?

No — irrational and transcendental.

£1000 at 5% continuous for 3 years?

1000e^{0.15}\approx\pounds1161.83

Connections

Natural logarithm ln x — the inverse of $e^x$
Exponential function e^x and its derivative — why $\frac{d}{dx}e^x=e^x$
Compound interest — the financial origin of the limit
Differential equations dy/dx = ky — natural growth/decay solved by $e^{kt}$
Binomial theorem — used to derive the $\sum 1/k!$ series
Limits and indeterminate forms — the $1^\infty$ case

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Socho tumhare paas £1 hai aur ek magic bank bolta hai "1 saal me tumhara paisa double kar dunga" (100% interest). Agar interest sirf ek baar milta hai, toh £1 → £2. Lekin agar tum bolo "mujhe ye interest chote-chote tukdon me do, aur har tukda turant apna interest kamaye" — yani compounding zyada baar ho — toh thoda zyada paisa banega. Aur zyada baar compound karo, aur thoda zyada. Logic kehta hai "infinite baar karo toh infinite paisa," par aisa nahi hota! Ye ruk jaata hai lagbhag 2.71828 pe. Wahi special number hai $e$ .

Formal roop se: $e=\lim_{n\to\infty}(1+\frac1n)^n$ . Yahan $\frac1n$ ek tukde ka size hai aur power $n$ batati hai kitni baar compound hua. Yaad rakho ye ek $1^\infty$ indeterminate form hai — andar wala $1$ ki taraf jaata hai par power infinity ki taraf, dono aapas me fight karte hain aur balance $e$ pe aata hai. Isliye "andar 1 hai toh answer 1 hoga" wali galti mat karna.

Convergence kyun hoti hai? Kyunki $e = 1 + 1 + \frac1{2!} + \frac1{3!} + \cdots$ , aur ye factorial wale terms bahut tezi se chote hote jaate hain ( $0.5, 0.167, 0.042\ldots$ ), toh total bounded reh jaata hai (3 se kam). Isiliye paisa infinite nahi hota.

Ye number sirf paise ke liye nahi — bacteria growth, radioactive decay, temperature cooling — jahan bhi cheez apni hi size ke proportional badhti/ghatti hai, wahan $e^{kt}$ aata hai. Isiliye ise "natural" growth kehte hain. Exam me formula $A=Pe^{rt}$ (continuous compounding) aur $N=N_0e^{kt}$ (growth) yaad rakhna — dono isi limit se aate hain.

The number e — definition as limit of (1+1 - n)ⁿ, natural growth context

What $e$ is

Deriving it from scratch (compound interest)

Why does it converge instead of blowing up?

Worked examples

Common mistakes

Active recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Exponentials & Logarithms

Connections