Term-by-term differentiation and integration of power series: Foundations

4.3.15 · D1 · Maths › Calculus III — Sequences & Series › Term-by-term differentiation and integration of power series

Yeh page kuch bhi assume nahi karta. Har letter, symbol aur picture jo parent note parent topic use karta hai, woh sab yahan ground up se build kiya gaya hai, ek aise order mein jahan har block pehle wale par tika ho.

0. Woh alphabet jo hum use karne wale hain

Kisi bhi formula se pehle, yahan har symbol ka plain-words mein matlab diya gaya hai. Hum neeche apne apne section mein har ek ko properly earn karte hain.

Symbol	Plain words	Section
$\sum$	"poori list ko add kar do"	§1
$n$	ek counter: $0, 1, 2, 3, \dots$	§1
$t_{n}$	ek generic term — list ka $n$ -va item	§1
$\infty$	"hamesha chalte rehna" — list kabhi khatam nahi hoti	§1
$a_{n}$	$n$ -ve piece par number weight	§2
$x$	woh input jo hum plug in karte hain	§2
$f$ , $f (x)$	ek machine jo input $x$ ko ek output number mein badal deti hai	§2
$(x - c)^{n}$	$n$ -va building block, $c$ par centred	§3
$c$	series ka centre point	§3
$∣ \cdot ∣$	absolute value: ek number ka size, sign hata ke	§4
$R$	radius: safe zone ki half-width	§4
$lim sup$	"woh sabse bada value jis par ek wobbly list baar baar laut ke aati hai"	§5
$\frac{d}{d x}, f^{'}$	slope machine (derivative)	§6
$\int$	area machine (integral)	§7
$C$	integration ka unknown constant	§7

1. $\sum$ , counter $n$ , aur $\infty$ — "ek list add karo"

Ise zor se padho: " $n$ ko $0$ se $3$ tak chalao, har $t_{n}$ ko add karte jao." Har $t_{n}$ ek term hai — shopping list ka ek item.

Figure s01 dekho. Cyan dots $\frac{1}{2} + \frac{1}{4} + \dots$ ke $1, 2, 3, \dots$ terms ke baad running total hain. White arrows dikhate hain ki har dot amber dashed line ki taraf creep kar raha hai jo height $1$ par hai — "homing in" (convergence) ki ek picture.

Figure — Term-by-term differentiation and integration of power series

2. Weights $a_{n}$ , input $x$ , aur function $f$

Ab hum har term ko ek number $x$ par dependent banate hain jo hum choose kar sakte hain.

Ek simple example: agar har $a_{n} = 1$ ho, toh terms hain $1, x, x^{2}, x^{3}, \dots$ aur $f (x) = \sum_{n = 0}^{\infty} x^{n} = 1 + x + x^{2} + x^{3} + \dots$ Yeh Geometric series hai — sabse friendly power series, aur parent note sab kuch isi se build karta hai.

3. Building block $(x - c)^{n}$ aur centre $c$

Toh parent note mein general power series hai $f (x) = \sum_{n = 0}^{\infty} a_{n} (x - c)^{n} = a_{0} + a_{1} (x - c) + a_{2} (x - c)^{2} + \dots$

jahan $f (x)$ (§2 se) woh number hai jis par poora sum land karta hai jab tum $x$ feed in karte ho.

Figure s02 dekho. Amber curve true total $\frac{1}{1 - x}$ hai; cyan/white curves woh hain jo tum $1 + x + x^{2} + \dots$ ke kuch terms ke baad rokne par paate ho. Har extra term amber curve ko ek wider stretch par hug karta hai — ek infinite polynomial, limit mein, wahi smooth function hai.

4. Absolute value $∣ \cdot ∣$ aur radius $R$

Ek bada $x$ plug karo aur powers $(x - c)^{n}$ explode karte hain; running totals infinity ki taraf bhaag jaati hain (diverge). Ek chhota $x$ plug karo aur terms shrink karti hain; running totals settle karti hain (converge). Ek cutoff distance hoti hai — aur distance direction ignore karta hai, aur yahi kaam absolute value ka hai.

Figure s03 dekho. Cyan band amber centre dot $c$ ke around half-width $R$ ka safe zone hai; white squares do edges $c \pm R$ mark karte hain, "test by hand" flagged ke saath.

5. $lim sup$ — coefficients se radius padhna

Hum $R$ find kaise karte hain? Cauchy–Hadamard theorem yeh akele weights se karta hai: $\frac{1}{R} = lim sup_{n \to \infty} ∣ a_{n} ∣^{1/ n} .$

Yahan do nayi cheezein hain.

Figure s04 dekho. Cyan dots $∣ a_{n} ∣^{1/ n}$ values ki ek wobbly list hain; amber dashed line $lim sup$ hai — woh ceiling jis par peaks baar baar laut ke aati hain chahe individual dots usse neeche dip kar lein.

Parent ka poora radius proof ek fact par tika hai: counter $n$ se weights multiply karna is $n$ -th-root scale ko barely nudge karta hai, kyunki $n^{1/ n} \to 1$ . Hum us number ko sanity-check kar sakte hain: $n = 100$ par bhi, $10 0^{1/100} \approx 1.047$ — already almost $1$ .

6. $\frac{d}{d x}$ aur $f^{'}$ — slope machine

Dekho woh rule ek term ke saath kya karta hai: woh weight ko $n$ se multiply karta hai aur power ko ek se giraata hai. Yahi wajah hai ki parent ka differentiated series $\sum n a_{n} (x - c)^{n - 1}$ padhta hai — har term wahi rule se hit hui. $n = 0$ term (ek constant) ki slope zero hoti hai aur woh vanish ho jaati hai, isliye sum $n = 1$ par restart karta hai.

7. $\int$ — area machine

$+ C$ (constant of integration) ek unknown constant hai: bahut saari functions ek hi slope-pattern share karti hain, sirf ek fixed vertical shift se differ karti hain, toh integration woh shift undetermined chhod deta hai jab tak tum use ek known value se pin nahi karte (jaise $f (0)$ ). Yahi wajah hai ki parent $C$ ko $arctan 0 = 0$ aur $ln (1 + 0) = 0$ use karke fix karta hai.

Har term par apply karne par, yeh parent ka integrated series $C + \sum \frac{a _{n}}{n + 1} (x - c)^{n + 1}$ deta hai: har weight $n + 1$ se divided, har power ek se bumped up.

Prerequisite map

Equipment checklist

Khud test karo — reveal karne se pehle jawab bolo.

n = 0 \sum \infty t_{n}

ka plain words mein kya matlab hai?

Never-ending list

t_{0} + t_{1} + t_{2} + \dots

ko add karo; yeh phir bhi ek finite number par land kar sakta hai agar terms kaafi tezi se shrink karein.

Ek series ke converge karne ka precisely kya matlab hai?

Uske running totals

s_{N}

N

badhne par ek fixed number

S

ke paas aate hain aur rehte hain; agar woh kabhi ek single number par settle nahi karte, toh diverge karta hai.

Polynomial kya hai, aur power series kaise alag hai?

Polynomial

x

ke whole-number powers ka ek finite sum hai fixed weights ke saath (e.g.

3 x^{2} + 5 x + 7

); power series wahi idea hai bina last term ke.

a_{n} (x - c)^{n}

mein kaunsa part pehle se fixed hai aur kaunsa tum choose karte ho?

a_{n}

(weight) aur

c

(centre) fixed hain;

x

woh input hai jo tum plug in karte ho.

∣ x - c ∣

kya measure karta hai, aur absolute value kyun?

x

c

tak ki distance; absolute value sign strip kar deta hai toh sirf size (distance) matter karta hai, direction nahi.

Radius of convergence

R

geometrically kya hai, aur

x = c

par kya hota hai?

Safe interval

(c - R, c + R)

ki half-width; series hamesha centre

x = c

par converge karti hai (wahan

a_{0}

hoti hai), chahe

R = 0

hi kyun na ho.

Plain limit ki jagah

lim sup ∣ a_{n} ∣^{1/ n}

kyun use karte hain?

Kyunki

∣ a_{n} ∣^{1/ n}

hamesha ke liye wobble kar sakta hai aur plain limit exist nahi kar sakta;

lim sup

ceiling hamesha exist karta hai aur pinpoint karta hai ki series pehli baar kahan break hoti hai.

\frac{d}{d x}

aur

\int

ke one-block rules batao.

\frac{d}{d x} x^{n} = n x^{n - 1}

(power down, times

n

);

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

(power up,

n + 1

se divide).

Integrate karne ke baad constant

+ C

kyun aata hai?

Bahut saari functions ek hi slope-pattern share karti hain, vertical shift se differ karti hain;

C

unknown rehta hai jab tak use

f (0)

jaisi known value se fix na karo.

Woh ek limit kaunsa hai jo dono radius proofs mein kaam aata hai?

n^{1/ n} \to 1

(aur isi tarah

(n + 1)^{\pm 1/ n} \to 1

), toh weights ko

n

se multiply/divide karna

n

-th-root scale ko unchanged chhod deta hai.