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

3.2.3 · Maths › Exponentials & Logarithms

$e$ kya hai

KYA matlab hai har piece ka:

$\frac{1}{n}$ = har interest chunk ka size (agar $n$ baar per year 100% par compound kar rahe ho).
Power $n$ = kitni baar compound karte hain.
$n \to \infty$ lena = continuously compound karna (har waqt).

Scratch se derive karna (compound interest)

KAISE growth build up hoti hai. 1 unit se shuru karo, rate $x$ per year par $n$ baar compound karo:

$after 1 chunk = 1 \cdot (1 + \frac{x}{n})$ $after 2 chunks = (1 + \frac{x}{n})^{2} (har chunk phir se multiply karta hai)$ $after all n chunks = (1 + \frac{x}{n})^{n}$

Yeh step kyun? Kyunki compounding repeated multiplication hai, addition nahi — yahi toh "interest on interest" ka poora point hai.

Ab continuous growth model karne ke liye $n \to \infty$ lene do. Yeh limit $e^{x}$ define karti hai.

Series ki derivation (Kyun?): Binomial theorem se, $(1 + \frac{1}{n})^{n} = \sum_{k = 0}^{n} (k n) \frac{1}{n ^{k}} = \sum_{k = 0}^{n} \frac{n ( n - 1 ) \dots ( n - k + 1 )}{k ! n ^{k}} .$ Fraction $\frac{n ( n - 1 ) \dots ( n - k + 1 )}{n ^{k}} = 1 \cdot (1 - \frac{1}{n}) (1 - \frac{2}{n}) \dots \to 1$ jab $n \to \infty$ . Yeh step kyun? Har factor $(1 - \frac{j}{n}) \to 1$ , isliye term $k$ , $\frac{1}{k !}$ ki taraf tend karta hai. Sum karne par $\sum \frac{1}{k !}$ milta hai. Yeh series fast converge karti hai, jo humein $e$ actually compute karne ka tarika deti hai.

Yeh converge kyun karta hai, blow up kyun nahi karta?

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

Worked examples

Worked example Example 1 — Limit ko converge hote dekhna

Kuch $n$ ke liye $(1 + \frac{1}{n})^{n}$ compute karo.

$n$	$(1 + \frac{1}{n})^{n}$
1	$2.000$
2	$2.250$
10	$2.594$
100	$2.705$
1000	$2.717$

Yeh step kyun? Concrete $n$ plug karne se dikhta hai sequence $2.71828$ ki taraf badhti hai lekin slow hoti jaati hai — definition action mein, sirf symbols nahi.

Common mistakes

Active recall

Recall Khud ko test karo (koshish ke baad kholo)

$e$ ko limit ki tarah define karo. → $e = lim_{n \to \infty} (1 + 1/ n)^{n}$ .
Yeh diverge kyun nahi karta? → Correction terms $\frac{1}{k !}$ super-fast shrink karte hain; 3 se upar bounded hai.
Limit kaunsa form hai? → $1^{\infty}$ indeterminate.
$e^{x}$ kaise milta hai? → $lim_{n \to \infty} (1 + x / n)^{n}$ .

Recall Feynman: ek 12-saal ke bachche ko samjhao

Tumne £1 ek magic bank mein daala jo waada karta hai ek saal mein tumhara paisa double kar dega. Lekin tum lalchi ho: poora saal wait karne ki bajaye, 6 mahine baad aadhi growth maangto, phir woh bhi grow karne do. Thoda £2 se zyada milta hai. Monthly bits mein maango — thoda aur. Har second, har instant karo... tumhe lagta bada paisa milega, lekin yeh roughly £2.72 par ruk jaata hai. Woh special rukne wala number $e$ kehlata hai. Yahi hai kitni tez cheezein grow karti hain jab woh "jitna smoothly ho sake" grow karti hain.

Flashcards

What is the definition of

e

as a limit?

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

(1 + \frac{1}{n})^{n}

sirf 1 kyun nahi hai?

Yeh

1^{\infty}

indeterminate form hai — base

\to 1

lekin exponent

\to \infty

; dono

e

par balance karte hain.

e

ki series expansion do.

e = \sum_{k = 0}^{\infty} \frac{1}{k !} = 1 + 1 + \frac{1}{2 !} + \frac{1}{3 !} + \dots

Series/limit converge kyun karta hai?

Terms

\frac{1}{k !}

geometrically se bhi tez shrink karte hain; sum 3 se upar bounded hai.

e^{x}

ka limit formula?

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

Continuous compound interest formula aur

e

kahan se aata hai?

A = P e^{r t}

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

ki limit jab

n \to \infty

Kya

e

rational hai?

Nahi — irrational aur transcendental.

£1000 at 5% continuous for 3 years?

1000 e^{0.15} \approx £1161.83

Connections

Natural logarithm ln x — $e^{x}$ ka inverse
Exponential function e^x and its derivative — kyun $\frac{d}{d x} e^{x} = e^{x}$
Compound interest — limit ka financial origin
Differential equations dy/dx = ky — natural growth/decay $e^{k t}$ se solve hota hai
Binomial theorem — $\sum 1/ k!$ series derive karne mein use hota hai
Limits and indeterminate forms — $1^{\infty}$ case