Taylor series — derivation from power series: Visual walkthrough

4.3.16 · D2 · Maths › Calculus III — Sequences & Series › Taylor series — derivation from power series

Step 1 — Ek polynomial actually hoti kya hai, picture mein

Kuch bhi infinite karne se pehle, aao agree karein ki pieces kya hain.

KYA HAI. Ek polynomial power terms ka sum hoti hai. Ek power term ek coefficient (ek fixed number) hoti hai jo variable ki power se multiply hoti hai: $fixed number c_{2} \cdot power of the displacement (x - a)^{2}$

$x$ woh input hai jo hum slide karte hain.
$a$ ek fixed point hai jise hum centre kehte hain — woh jagah jahan hum apna approximation banate hain.
$(x - a)$ displacement hai: $x$ ne centre se kitni door move ki hai. Yeh exactly $0$ hota hai jab $x = a$ .
$c_{n}$ coefficient hai — ek dial jise hum curve ko shape karne ke liye adjust kar sakte hain.

YAHAN SE KYUN SHURU KAREIN. Poori derivation har dial $c_{n}$ ki sahi value dhundhne ke baare mein hai. Toh pehle hume dekhna hoga ki har dial physically kya control karti hai.

PICTURE. Neeche, wahi centre $a$ fixed hai. Har term $c_{n} (x - a)^{n}$ apne aap draw ki gayi hai: ek flat line ( $n = 0$ ), ek seedha ramp ( $n = 1$ ), ek parabola ( $n = 2$ ), ek cubic ( $n = 3$ ). Dhyan do ki sab $x = a$ par zero-height se guzarti hain sivaaye flat wali ke. Yahi ek fact poore proof ka beej hai.

Figure — Taylor series — derivation from power series

Step 2 — Woh ek move jo displacement ko gayab kar deta hai: $x = a$ set karo

KYA HAI. Centre khud, $x = a$ , ko series mein substitute karo.

KYUN YAHI MOVE AUR KOI NAHI? Koi bhi term dekho jahan $n \geq 1$ ho: usme factor $(x - a)$ hai. $x = a$ par woh factor $(a - a) = 0$ hota hai, aur $0$ times kuch bhi $0$ hota hai. Toh $x = a$ substitute karna ek term-killer hai: yeh har term ko delete kar deta hai sivaaye flat wali, $c_{0}$ , ke. Hum woh ek input value choose kar rahe hain jo ek single survivor ko chhod kar baaki sabko chup kara deta hai.

$f (a) = survives c_{0} + = 0 c_{1} \cdot 0 + = 0 c_{2} \cdot 0 + \dots = c_{0}$

PICTURE. Dekho vertical line $x = a$ stacked terms se guzar rahi hai. Har curved/ramped term exactly apne zero-height crossing par pakdi jaati hai; sirf flat line $c_{0}$ ki koi height bachi rehti hai. Woh height hi $f (a)$ hai.

Step 3 — Hume ek NAYE tool ki zaroorat kyun hai: derivative

Hamare paas $c_{0}$ hai. Lekin $x = a$ dobara set karne se phir $c_{0}$ hi milega — hum kuch naya nahi seekhte. Hum stuck hain.

KYA HAI. Hum derivative $f^{'} (x)$ introduce karte hain: har point par curve ki slope (steepness).

DERIVATIVE HI KYUN? Hume ek aisi machine chahiye jo (a) constant $c_{0}$ ko delete kare taaki woh hamara view block karna band kare, aur (b) agli dial $c_{1}$ ko "flat" slot mein neeche kheeche taaki $x = a$ set karna use pakad sake. Differentiation exactly dono karta hai:

ek constant ki derivative $0$ hoti hai ( $c_{0}$ delete karta hai),
$c_{1} (x - a)$ ki derivative $c_{1}$ hai (ramp ko ek naye flat line mein badalta hai).

Yeh akela elementary tool hai jo har power ko ek se neeche karta hai — $n = 1$ ki picture akele dekhne ke liye Linear approximation & differentials dekho.

$f^{'} (x) = from c_{0} 0 + ramp \to flat c_{1} + still has (x - a) 2 c_{2} (x - a) + still has (x - a) 3 c_{3} (x - a)^{2} + \dots$

PICTURE. Top panel mein ramp $c_{1} (x - a)$ upar ki taraf tilt kar raha hai; bottom panel mein uski derivative hai — height $c_{1}$ par ek flat line. Derivative "steepness padhti hai" aur use constant ki tarah report karti hai.

Step 4 — Term-killer ko dobara chalao: $f^{'}$ mein $x = a$ set karo

KYA HAI. Ab $f^{'} (x)$ mein $x = a$ substitute karo.

KYUN. Same logic Step 2 jaisi, ek level upar. $f^{'}$ mein har term jo abhi bhi $(x - a)$ carry karti hai woh mar jaati hai; sirf freshly-flattened $c_{1}$ bachta hai.

$f^{'} (a) = c_{1} + all killed 2 c_{2} \cdot 0 + 3 c_{3} \cdot 0 + \dots = c_{1}$

PICTURE. Vertical line $x = a$ derivative graph ko slice karti hai. Saare curved leftovers zero-height par pakde jaate hain; sirf flat $c_{1}$ line ki height hai. Woh height $f^{'} (a)$ hai, centre par $f$ ki slope.

Ab hum rhythm dekh rahe hain: agli term ko flatten karne ke liye differentiate karo, phir use pakadne ke liye $x = a$ set karo. Repeat karo.

Step 5 — Do baar differentiate karo: jahan factorial paida hota hai

KYA HAI. $c_{2}$ tak pahunchne ke liye dobara differentiate karo. Woh number dekho jo quadratic term se nikalta hai.

WATH NUMBER KO KYUN DEKHO? Yahi poore page ka core hai. Term $c_{2} (x - a)^{2}$ ek step mein flat nahi hoti — ise do steps chahiye. Aur har step current exponent se multiply karti hai:

$c_{2} (x - a)^{2} \frac{d}{d x} 2 c_{2} (x - a)^{1} \frac{d}{d x} 2 \cdot 1 c_{2} = 2! c_{2}$

Pehli derivative exponent $2$ peel karti hai.
Doosri derivative naya exponent $1$ peel karti hai.
Unka product $2 \cdot 1 = 2! = 2$ , $c_{2}$ ke aage aa jaata hai.

Toh do derivatives ke baad: $f^{''} (a) = 2! c_{2} \Rightarrow c_{2} = \frac{f ^{''} ( a )}{2 !}$

PICTURE. Ek "peeling staircase": upar parabola $(x - a)^{2}$ , beech mein uski pehli derivative $2 (x - a)$ (ab " $2$ " slope ke roop mein visible), neeche doosri derivative $2$ (height $2$ par flat line). Girte exponents $2$ phir $1$ laal rang mein label kiye gaye hain jaise woh neeche aate hain.

Step 6 — General term: $n$ baar differentiate karo, $n! c_{n}$ pakdo

KYA HAI. General term $c_{n} (x - a)^{n}$ ke liye peeling karo.

KYUN. Hum ek aisa formula chahte hain jo ek saath har dial ko cover kare, toh hum saare peeled exponents track karte hain.

Har differentiation exponent ko ek se ghata deti hai aur purane exponent se multiply karti hai: $(x - a)^{n} \to n (x - a)^{n - 1} \to n (n - 1) (x - a)^{n - 2} \to \dots \to = n! n (n - 1) (n - 2) \dots 1$

Exactly $n$ derivatives ke baad, saare $n$ exponents $n, n - 1, \dots, 1$ peel ho chuke hain aur aapas mein multiply ho gaye hain.
Unka product factorial hai $n! = n \cdot (n - 1) \dots 2 \cdot 1$ .
Lower terms (powers $< n$ ) differentiate hoke $0$ ho gayi hain aur gayab ho gayi hain; higher terms abhi bhi $(x - a)$ carry karti hain aur $x = a$ se kill ho jayengi.

Akele survivor ko pakadne ke liye $x = a$ set karo: $f^{(n)} (a) = n! c_{n} \Rightarrow c_{n} = \frac{f ^{(n)} ( a )}{n !}$

Yahan $f^{(n)}$ ka matlab hai " $f$ ko $n$ baar differentiate kiya gaya."

PICTURE. Girte exponents ki ek ladder: $n \to n - 1 \to \dots \to 1$ , har rung us factor ke saath label kiya gaya jo woh contribute karta hai, aur running product $n!$ daayein taraf accumulate hota hua. Neeche division bar $\div n!$ dikhata hai ki factorial dial se cancel ho raha hai.

Step 7 — Edge case: agar $f^{(n)} (a) = 0$ ho toh?

KYA HAI. Maan lo koi derivative centre par zero hai, jaise $cos x$ ke liye $a = 0$ par saari odd derivatives $0$ deti hain.

KYUN COVER KAREIN. Ek dial ka zero hona koi error nahi hai — iska matlab hai woh term simply absent hai, aur curve ki us order mein centre par koi ramp/kink nahi hai. Formula ise automatically handle karta hai: $c_{n} = 0/ n! = 0$ .

$cos x : f^{'} (0) = 0, f^{'''} (0) = 0, \dots \Rightarrow c_{1} = c_{3} = \dots = 0$ $cos x = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots (only even powers)$

PICTURE. $cos x$ , $0$ ke paas perfectly flat-topped aur symmetric hai — koi left/right tilt nahi — toh koi bhi odd (asymmetric) term allowed nahi hai. Even terms symmetric dome banate hain; odd dials switch off hain.

Step 8 — Edge case: centre zero par nahi ( $a \neq = 0$ )

KYA HAI. Peeling ko dobara karo jab centre, maan lo, $ln x$ ke liye $a = 1$ ho.

KYUN. Killer move ko factor $(x - a)$ ka centre par vanish hona chahiye. Agar hum laparwahi se $(x - 1)$ ki jagah $x$ ki powers likhein, toh $x = 1$ plug karne se woh zero nahi hongi, aur poora term-killing machine kharab ho jaayega. Basis zaroor $(x - a)^{n}$ hona chahiye.

$f (x) = ln x : f (1) = 0, f^{'} (1) = 1, f^{''} (1) = - 1, f^{'''} (1) = 2$ $c_{0} = 0, c_{1} = \frac{1}{1 !} = 1, c_{2} = \frac{- 1}{2 !} = - \frac{1}{2}, c_{3} = \frac{2}{3 !} = \frac{1}{3}$ $ln x = (x - 1) - \frac{( x - 1 ) ^{2}}{2} + \frac{( x - 1 ) ^{3}}{3} - \dots$

PICTURE. Approximating polynomial $ln x$ ko $x = 1$ ke paas tight hug karta hai aur dur jaake drift ho jaata hai — accuracy centre par anchored hai, exactly wahan jahan $(x - a)$ tiny hota hai.

Ek-picture summary

Upar ki saari cheez ek single loop mein compress hoti hai, $n = 0, 1, 2, \dots$ ke liye repeat:

$flattens c_{n}, builds n! differentiate n times ⟶ kills all but one term set x = a ⟶ f^{(n)} (a) = n! c_{n} ⟶ c_{n} = \frac{f ^{(n)} ( a )}{n !}$

Recall Feynman: poori walkthrough ko plain words mein dobara sunao

Humne guess kiya ki hamari curvy function secretly ek giant polynomial hai jisme adjustable dials ki ek row hai. Pehli dial dhundhne ke liye, humne centre point plug in kiya — yeh waisa hi hai jaise slide ki height wahan padhna jahan tum khade ho; har wobbly part wahan momentarily flat hota hai, toh sirf base height dikhti hai. Agli dial dhundhne ke liye humne slope measure ki (derivative), jo base height erase kar deti hai aur ramp ko ek naye readable height mein badal deti hai — phir dobara centre plug in kiya. Dial number teen tak pahunchne ke liye humne slope kaise change hoti hai measure kiya (do baar differentiate), aur aise hi aage. Har baar differentiate karne par, powers "peel off" ho jaati hain jaise multiplying numbers jo $n!$ mein pile up ho jaate hain; toh hum $n!$ se divide karte hain taaki dials honest rahein. Agar koi dial zero aati hai, toh woh shape simply curve ka part nahi hai. Aur hum hamesha "centre se doori" ke powers measure karte hain, $(x - a)$ , taaki centre plug in karna sach mein un sabko chup kara sake sivaaye us ek dial ke jise hum dhundh rahe hain.

Connections

Parent: full algebraic derivation
Maclaurin series — common expansions — $a = 0$ ki pictures $e^{x}, sin, cos$ ke liye
Linear approximation & differentials — Steps 3–4 akele ( $n = 1$ )
Taylor's theorem with remainder — truncated picture $a$ se door kitna galat hai
Power series — radius & interval of convergence — yahan yeh pictures valid rehna band kar deti hain
Geometric series — woh special case jahan saari dials equal $1$ hain
L'Hôpital's rule — yeh series ki leading terms use replace kar deti hain