Taylor series approximation — Hinglish notes

1.2.7 · AI-ML › Calculus & Optimization Basics

WHAT hai Taylor series?

HOW derive karte hain? (scratch se)

Hum chahte hain ek polynomial $P (x) = c_{0} + c_{1} (x - a) + c_{2} (x - a)^{2} + \dots$ jo $x = a$ par $f$ ke saath jitna ho sake utna agree kare.

Step 1 — Value match karo. $x = a$ set karo: $(x - a)$ wale saare terms zero ho jaate hain, isliye $P (a) = c_{0}$ . Hum chahte hain $P (a) = f (a)$ , toh $c_{0} = f (a)$ . Yeh step kyon? Constant term $a$ par height control karta hai; koi doosra term usse touch nahi karta.

Step 2 — Slope match karo. Differentiate karo: $P^{'} (x) = c_{1} + 2 c_{2} (x - a) + 3 c_{3} (x - a)^{2} + \dots$ . $x = a$ set karo: $P^{'} (a) = c_{1}$ . Hum chahte hain $P^{'} (a) = f^{'} (a)$ , toh $c_{1} = f^{'} (a)$ . Yeh step kyon? Slope match karne se line $f$ ko tangentially touch karti hai, sirf cross nahi.

Step 3 — Curvature match karo. Dobara differentiate karo: $P^{''} (x) = 2 c_{2} + 6 c_{3} (x - a) + \dots$ . $x = a$ set karo: $P^{''} (a) = 2 c_{2}$ , toh $c_{2} = \frac{f ^{''} ( a )}{2}$ . Yeh step kyon? $2$ ka factor isliye aata hai kyunki $(x - a)^{2}$ ko do baar differentiate karne par $2$ milta hai. Yahi factorial ka seed hai.

Step 4 — Pattern dekho. $n$ baar differentiate karo: term $c_{n} (x - a)^{n}$ , $x = a$ par $n! c_{n}$ ban jaata hai (neeche ke saare terms khatam ho jaate hain, upar ke sabhi terms mein abhi bhi $(x - a)$ hota hai). Toh $f^{(n)} (a) = n! c_{n} \Rightarrow c_{n} = \frac{f ^{(n)} ( a )}{n !} .$ Yeh step kyon? $n!$ koi magic nahi hai — yeh exactly woh number hai jo $x^{n}$ deta hai jab usse $n$ baar differentiate karo. Usse divide karna us factor ko "undo" karta hai.

Wapas plug karne par definition milti hai. Humne isse sirf yeh demand karke derive kiya ki saare derivatives match hon.

Worked examples

Common mistakes

Flashcards

n

-th Taylor coefficient kiske barabar hota hai?

\frac{f ^{(n)} ( a )}{n !}

—

a

par

n

-th derivative divided by

n!

Denominator mein factorial kyon hota hai?

(x - a)^{n}

ko exactly

n

baar differentiate karne par

n!

milta hai; usse divide karna us factor ko cancel karta hai taaki derivatives match hon.

Maclaurin series kya hoti hai?

a = 0

ke around expand ki gayi Taylor series.

e^{x}

ki Maclaurin series batao.

1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots

Truncation error

a

ke paas kaise scale karta hai?

Order-

k

polynomial ke liye

(x - a)^{k + 1}

ki tarah (Lagrange remainder).

Kaun sa Taylor order gradient descent se correspond karta hai?

First order:

L (θ_{0} + Δ) \approx L (θ_{0}) + \nabla L^{⊤} Δ

Kaun sa order Newton's method se correspond karta hai?

Second order, step deta hai

Δ = - H^{- 1} \nabla L

Kya Taylor series hamesha apne function ke barabar hoti hai?

Nahi — sirf apne radius of convergence ke andar, aur kuch smooth functions (jaise

e^{- 1/ x^{2}}

) apni series ke barabar bilkul nahi hote.

Multivariable 2nd-order Taylor form?

f (x_{0} + Δ) \approx f (x_{0}) + \nabla f^{⊤} Δ + \frac{1}{2} Δ^{⊤} H Δ

Recall Ek 12-saal ke bachhe ko samjhao (Feynman)

Socho tumhe pata hai ek toy car bilkul kahan hai, kitni tez ja rahi hai, aur kitni tez speed up ho rahi hai — sab ek hi moment mein. Sirf un teen facts se tum andaza laga sakte ho ki wo thodi der baad kahan hogi. Taylor series kisi bhi curve ke saath yahi karta hai: yeh ek jagah ki height, slope, bend, twist... use karta hai us curve ki ek copy draw karne ke liye jo paas mein ho. Jitne zyada facts (derivatives) use karo, copy utni better — lekin sirf un jagahon ke liye jo jahan se tumne shuru kiya wahan ke kareebi hain. Door jaane par tumhara andaza toot jaata hai.

Connections

Derivatives and gradients — Taylor poori tarah derivatives se bana hai.
Gradient Descent — 1st-order Taylor truncation.
Newton's Method — 2nd-order Taylor truncation.
Hessian matrix — multivariable Taylor ka curvature term.
Convex functions — $H ⪰ 0$ ke saath 2nd-order Taylor convexity imply karta hai.
Approximation error and Big-O — remainder term accuracy ko formalize karta hai.