Series — partial sums, convergence definition: Foundations

4.3.3 · D1 · Maths › Calculus III — Sequences & Series › Series — partial sums, convergence definition

Parent note partial sums aur convergence padhne se pehle, tumhare paas har woh symbol hona chahiye jo woh tumhare upar fire karta hai. Yeh page unhe us order mein list karta hai jismein woh ek-dusre par build karte hain — kuch bhi apni definition se pehle nahi aata, aur har cheez ek picture se anchored hai.

1. Subscript $a_{n}$ — ek numbered slot

Picture: ek row of numbered mailboxes imagine karo. Box $1$ mein ek number hai, box $2$ mein ek number hai, aur yahi silsila forever chalta hai. $a_{n}$ hai "jo bhi number box $n$ mein rehta hai."

Figure — Series — partial sums, convergence definition

Topic ko yeh kyun chahiye. Ek series numbers ki ek specific list add karti hai. "General term" ke baare mein baat karne ke liye bina sab likh diye, humein ek symbol chahiye, $a_{n}$ , jo kisi bhi address $n$ par entry ko represent kare.

2. Ek sequence $(a_{n})$ — poori numbered row

Picture: wohi mailboxes ki row, lekin ab tum peeche hato aur poori row ko ek saath ek single cheez ki tarah dekho, ek ek box ki jagah. Row ka ek shape hai — shayad numbers $0$ ki taraf shrink ho jaate hain, shayad badhte hain, shayad wobble karte hain.

Topic ko yeh kyun chahiye. Parent note ka punchline hai "ek series disguise mein ek sequence HAI." Jab tak sequence ka matlab "ek object jo infinitely many ordered numbers se bana ho" nahi ho jaata, tab tak tum yeh nahi samajh sakte. Poori kahani ke liye Sequences — limits and convergence dekho.

3. Limit $lim_{n \to \infty} a_{n} = L$ — row kahan ja rahi hai

" $\to \infty$ " yahan kyun, kisi finite stop par kyun nahi? Kyunki hum long-run behaviour ke baare mein pooch rahe hain. Koi bhi finite address special nahi hai; sawaal yeh hai ki agar tum kabhi chalna band nahi karte toh numbers kahan settle hote hain. Arrow $\to$ ko "approaches" padho.

Topic ko yeh kyun chahiye. Ek series ki convergence defined hai ek limit ke roop mein — lekin running totals ke limit ke roop mein, terms ke nahi. Inhe alag rakhna hi poora game hai (Parent note mein Mistake 2).

4. Summation sign $\sum$ — "inhe add karo"

$\sum_{n = 1}^{4} a_{n} = a_{1} + a_{2} + a_{3} + a_{4} .$

Picture: ek machine jisme ek start dial ( $n = 1$ ) aur ek stop dial ( $N = 4$ ) hai. Woh ek ek mailbox uthati hai, start se stop tak, aur unke saare numbers ek running bucket mein daalti hai.

Top number kyun matter karta hai. Agar stop dial finite $N$ dikhata hai, toh tumhe genuinely ek finite sum milta hai — tum kar sakte ho. Agar top $\infty$ dikhata hai, toh tum literally forever add nahi kar sakte; yahi exactly woh problem hai jo topic partial sums dekh kar solve karta hai.

5. Partial sum $s_{N}$ — ek running total jo ruk jaata hai

Picture: apne bucket ke paas khade raho. $s_{1}$ = box 1 dalo. $s_{2}$ = box 2 bhi dalo. $s_{3}$ = box 3 bhi. Har ek pour ke baad bucket ka level ek partial sum hai. Level ko pour-by-pour badhte dekhna sequence $(s_{N})$ ko dekhna hai.

Topic ko yeh kyun chahiye — central trick. "Infinitely many numbers add karo" undefined hai (addition kabhi bhi sirf do cheezein ek saath combine karti hai). Lekin $s_{1}, s_{2}, s_{3}, \dots$ ek bilkul ordinary sequence hai, aur ek sequence ya toh limit rakhti hai ya nahi. Toh topic infinite sum ko redefine karta hai: $\sum_{n = 1}^{\infty} a_{n} = lim_{N \to \infty} s_{N} .$ Impossible "forever add karo" ban jaata hai jaana-pehchana "kya yeh sequence converge karti hai?"

6. Converge / diverge — kya bucket level settle hota hai?

Picture, dono cases:

Converge: bucket level badhta hai lekin dheema ho jaata hai, ek fixed line $S$ se chipka rehta hai jise woh kabhi cross nahi karta.
Diverge: level ya toh bina ceiling ke badhta rehta hai, ya up-down-up-down karata rehta hai aur kabhi ghar nahi dhundta.

7. Quantifiers $\forall$ , $\exists$ aur $ε$ — precise "settles"

Inhe mila kar, $lim_{N \to \infty} s_{N} = S$ ka matlab hai: $\forall ε > 0, \exists M s.t. N \geq M ⟹ ∣ s_{N} - S ∣ < ε .$

Plain words: koi bhi tolerance band chuno $S$ ke aaspaas half-width $ε$ ka (chahe kitna bhi patla ho). Ek cutoff $M$ hai jiske baad har running total us band ke andar aata hai aur rehta hai.

Itni precision kyun chahiye. "Gets closer and closer" vague hai — kitni tezi se close, kiske closer? $ε$ -band "settles" ko testable banata hai: chahe ek skeptic kitna bhi patla band kheenche, tum ek aisa point naam le sakte ho jiske baad tum kabhi ushe nahi chhodte.

Yeh topic ko kaise feed karte hain

Ise upar se padho: ek single term $a_{n}$ ek sequence banata hai; $\sum$ se ek finite range add karna ek partial sum $s_{N}$ banata hai; partial sums apna khud ka sequence banate hain; yeh poochna ki kya woh sequence ek limit rakhti hai — $ε$ se precise kiya gaya — exactly yahi hai jiska matlab hai ki series converge karti hai.

Har foundation kahan dobara aata hai

Foundation	Topic mein pehli baar istamal...
$a_{n}$ , sequence	Geometric series ka general term likhne mein
limit of $(a_{n})$	n-th Term Test for Divergence ( $a_{n} \to 0$ ?) mein
$\sum$ , $s_{N}$	har closed-form partial sum derive karne mein
telescoping cancellation	Telescoping series mein
$a_{n} \to 0$ lekin $\sum$ diverge karta hai	Harmonic series mein

Equipment checklist

Khud test karo — right side cover karo.

a_{3}

ka kya matlab hai, aur kya

3

ek multiplier hai?

List ka teesra term;

3

ek address/label hai, multiplication nahi.

Ek sequence

(a_{n})

kya hai (ek single term ke versus)?

Poori ordered infinite list

a_{1}, a_{2}, a_{3}, \dots

ek object ki tarah li gayi.

lim_{n \to \infty} a_{n} = L

plain words mein kya kehta hai?

Jaise addresses badhte hain, terms

L

ke arbitrarily close aa jaate hain aur rehte hain.

\sum_{n = 1}^{4} a_{n}

zor se padho aur expand karo.

a_{n}

ka sum

n = 1

4

tak"

= a_{1} + a_{2} + a_{3} + a_{4}

\sum_{n = 1}^{\infty} a_{n}

ko direct addition se kyun compute nahi kar sakte?

Addition kabhi bhi sirf do numbers combine karti hai; tum infinitely many steps perform nahi kar sakte.

Partial sum

s_{N}

define karo.

s_{N} = a_{1} + a_{2} + \dots + a_{N} = \sum_{n = 1}^{N} a_{n}

Convergence decide karne ke liye hum actually kaun sa sequence dekhte hain?

Partial sums ka sequence

(s_{N})

, terms ka sequence

(a_{n})

nahi.

\sum a_{n}

converges to

S

ka exactly kya matlab hai?

lim_{N \to \infty} s_{N} = S

, ek finite number.

Kya "diverges" ka hamesha matlab "infinity ki taraf jaata hai" hota hai?

Nahi — iska matlab hai koi finite limit nahi; yeh forever bounce bhi kar sakta hai, jaise

- 1, 0, - 1, 0, \dots

\forall

aur

\exists

ka kya matlab hai?

"For every" aur "there exists."

∣ s_{N} - S ∣

kya measure karta hai?

Running total aur target

S

ke beech distance (sign ignored).

Connections

Parent topic ↑
Sequences — limits and convergence (ek series partial sums ka sequence hai)
Geometric series · Telescoping series · Harmonic series
n-th Term Test for Divergence