Maclaurin series of eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ — derive all

4.3.17 · Maths › Calculus III — Sequences & Series

1. Master formula (ek baar derive karo, hamesha use karo)

Scratch se Derivation. Maano $f$ ko ek power series ke roop mein likha ja sakta hai: $f (x) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots$

$a_{0}$ dhundho: $x = 0$ rakho. $x$ wale har term mar jaate hain → $f (0) = a_{0}$ . Kyun? $0^{k} = 0$ .
$a_{1}$ dhundho: ek baar differentiate karo: $f^{'} (x) = a_{1} + 2 a_{2} x + 3 a_{3} x^{2} + \dots$ , phir $x = 0$ rakho → $f^{'} (0) = a_{1}$ .
$a_{2}$ dhundho: dobara differentiate karo: $f^{''} (x) = 2 a_{2} + 6 a_{3} x + \dots$ , $x = 0$ rakho → $f^{''} (0) = 2 a_{2}$ , toh $a_{2} = \frac{f ^{''} ( 0 )}{2}$ .
General pattern: $n$ baar differentiate karne par, $x^{n}$ term $n! a_{n}$ ban jaata hai aur $x = 0$ par yahi akela bachta hai.

2. $e^{x}$ — self-replicating function

KAISE: $f^{(n)} (0) = e^{0} = 1$ sabhi $n$ ke liye, toh $a_{n} = \frac{1}{n !}$ . $e^{x} = n = 0 \sum \infty \frac{x ^{n}}{n !} = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots (all x)$

3. $sin x$ aur $cos x$ — cycling derivatives

$sin x$ ke liye KAISE: $f (0) = 0, f^{'} (0) = 1, f^{''} (0) = 0, f^{'''} (0) = - 1, \dots$ $sin x = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots = n = 0 \sum \infty \frac{( - 1 ) ^{n} x ^{2 n + 1}}{( 2 n + 1 )!}$

$cos x$ ke liye KAISE: $f (0) = 1, f^{'} (0) = 0, f^{''} (0) = - 1, \dots$ — sirf even powers: $cos x = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots = n = 0 \sum \infty \frac{( - 1 ) ^{n} x ^{2 n}}{( 2 n )!}$

Figure — Maclaurin series of eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ — derive all

4. $ln (1 + x)$ — geometric series integrate karke derive karo

KAISE. Geometric series ( $\sum r^{n} = \frac{1}{1 - r}$ se, $r = - x$ rakh ke): $\frac{1}{1 + x} = 1 - x + x^{2} - x^{3} + \dots (∣ x ∣ < 1)$ Term by term integrate karo, constant fix karne ke liye $ln (1 + 0) = 0$ use karo: $ln (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots = n = 1 \sum \infty \frac{( - 1 ) ^{n - 1} x ^{n}}{n} (- 1 < x \leq 1)$

5. $(1 + x)^{n}$ — Binomial Series

KAISE. $f (x) = (1 + x)^{n}$ . $f^{'} (x) = n (1 + x)^{n - 1}$ , $f^{''} (x) = n (n - 1) (1 + x)^{n - 2}$ , ..., $f^{(k)} (x) = n (n - 1) \dots (n - k + 1) (1 + x)^{n - k}$ . $x = 0$ par: $f^{(k)} (0) = n (n - 1) \dots (n - k + 1)$ . $k!$ se divide karo: $(1 + x)^{n} = k = 0 \sum \infty (k n) x^{k} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \dots (∣ x ∣ < 1 if n \in / Z_{\geq 0})$ jahan $(k n) = \frac{n ( n - 1 ) \dots ( n - k + 1 )}{k !}$ .

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tumhe sirf pata hai ki ek runner abhi kahan hai, plus uski speed, plus uski speed kitni tezi se badal rahi hai, aur aage bhi aisa hi. Itna kaafi hai unka poora path tumhare paas predict karne ke liye! Ek Maclaurin series curve ke liye yahi karta hai: ye $x = 0$ par height, slope, bend, bend-ka-bend copy karta hai, aur unhe $1, x, x^{2}, x^{3}, \dots$ ke pieces ke roop mein stack karta hai. Har piece ko ek factorial se divide kiya jaata hai taaki ye baaki sabko dabaye nahi. Kaafi pieces jodo aur tumhari simple polynomial real curve se chipak jaayegi.

Flashcards

Master Maclaurin coefficient formula

a_{n} = \frac{f ^{(n)} ( 0 )}{n !}

, toh

f (x) = \sum \frac{f ^{(n)} ( 0 )}{n !} x^{n}

n!

kyun aata hai

\frac{d ^{n}}{d x ^{n}} x^{n} = n!

x = 0

par akela bachne wala term

e^{x}

ki series

\sum_{n = 0}^{\infty} \frac{x ^{n}}{n !} = 1 + x + \frac{x ^{2}}{2 !} + \dots

, sabhi

x

ke liye valid

sin x

ki series

\sum \frac{( - 1 ) ^{n} x ^{2 n + 1}}{( 2 n + 1 )!} = x - \frac{x ^{3}}{6} + \frac{x ^{5}}{120} - \dots

(odd powers)

cos x

ki series

\sum \frac{( - 1 ) ^{n} x ^{2 n}}{( 2 n )!} = 1 - \frac{x ^{2}}{2} + \frac{x ^{4}}{24} - \dots

(even powers)

ln (1 + x)

ki series + convergence

x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots

, valid

- 1 < x \leq 1

ln (1 + x)

jaldi derive kaise karo

\frac{1}{1 + x} = 1 - x + x^{2} - \dots

ko term-by-term integrate karo

Binomial series

(1 + x)^{n}

\sum_{k} (k n) x^{k} = 1 + n x + \frac{n ( n - 1 )}{2} x^{2} + \dots

∣ x ∣ < 1

agar

n \in / Z_{\geq 0}

1 + x

ke pehle teen terms

1 + \frac{x}{2} - \frac{x ^{2}}{8} + \dots

sin

mein sirf odd powers kyun

sin

ek odd function hai ⇒ even-power coefficients zero ho jaate hain

Connections

Taylor series (general centre a) — Maclaurin $a = 0$ wala case hai
Geometric series — $ln (1 + x)$ aur $(1 + x)^{- 1}$ ki backbone
Radius & interval of convergence — ye series kahan valid hain
Euler's formula — $e^{i x} = cos x + i sin x$ teen series combine karke
L'Hôpital & limits via series — jaise $lim_{x \to 0} \frac{s i n x}{x} = 1$
Binomial theorem (integer n) — special terminating case