Sequences — convergence, divergence, boundedness, monotonicity

4.3.1 · Maths › Calculus III — Sequences & Series

WHAT is a sequence?

Pehle 5, 50, ya 5 million terms KABHI convergence decide nahi karte — sirf tail karta hai.

WHAT does convergence mean? (build the definition from scratch)

Hum kehna chahte hain " $a_{n}$ , $L$ ke paas aata hai aur paas rehta hai." Hum "paas" ko bina haath hilaye precise kaise banate hain?

WHY the $ε$ – $N$ definition? Kyunki "paas aata hai" vague hai. Toh hum ek challenger ko koi bhi tolerance $ε > 0$ choose karne dete hain ( $L$ ke around ek target band). Humein ek cutoff $N$ se respond karna hoga taaki $N$ ke baad ki har term us band ke andar ho. Agar hum hamesha respond kar sakte hain, chahe $ε$ kitna bhi tiny ho, toh terms sach mein $L$ par aa ke tik jaati hain.

Divergence — the flavours

Boundedness — ek necessary condition

Monotonicity — the second tool

HOW to test: $a_{n + 1} - a_{n}$ compute karo (sign?) ya ratio $a_{n + 1} / a_{n}$ vs $1$ (positive terms ke liye), ya $f (x)$ ko differentiate karo jahan $a_{n} = f (n)$ .

The payoff: Monotone Convergence Theorem (MCT)

Figure — Sequences — convergence, divergence, boundedness, monotonicity

Useful limit tools (taaki aapko hamesha $ε$ – $N$ use na karna pade)

Forecast-then-Verify drill

Common mistakes (Steel-manned)

Flashcards

Formally sequence kya hoti hai?

Ek function

a : N \to R

, likha jaata hai

(a_{n})

a_{n} \to L

ε

–

N

definition batao.

\forall ε > 0 \exists N : n > N \Rightarrow ∣ a_{n} - L ∣ < ε

Convergent ⟹ ?

Bounded (proof:

ε = 1

tail + finite head).

Kya bounded ⟹ convergent? Counterexample do.

Nahi;

a_{n} = (- 1)^{n}

bounded hai lekin oscillate karta hai.

Monotone Convergence Theorem batao.

Ek monotone bounded sequence converge karti hai (apne sup/inf par).

MCT ke neeche kaun sa axiom hai?

Completeness of

R

(least upper bound property).

Ek convergent recursive sequence

a_{n + 1} = f (a_{n})

ka limit kaise dhundhte hain?

Fixed point solve karo

L = f (L)

Sequences ke liye Squeeze Theorem batao.

Agar

b_{n} \leq a_{n} \leq c_{n}

aur

b_{n}, c_{n} \to L

toh

a_{n} \to L

(- 1)^{n}

diverge kyun karta hai (ek line mein)?

Tail width-

< 2

band mein fit nahi ho sakti jabki terms

2

distance apart rehti hain.

Kya pehle 1000 terms badalne se convergence change hoti hai?

Nahi — sirf tail matter karta hai.

(a_{n})

ki monotonicity test karo?

a_{n + 1} - a_{n}

ka sign, ya positive terms ke liye

a_{n + 1} / a_{n}

1

(1 + 1/ n)^{n}

ka limit?

e

(increasing + bounded above by 3 ⟹ MCT).

Recall Feynman: explain to a 12-year-old

Imagine karo tum ek wall ki taraf chal rahe ho aur har step mein tum baaki bachi distance ka aadha chalte ho. Tumhare footprints positions ki ek list banate hain: woh list ek sequence hai. Converging ka matlab hai tum wall ke aur aur paas aate rehte ho aur basically wahan ja ke ruk jaate ho. Diverging ka matlab hai ya tum forever bhaag jaate ho, ya tum baar baar aage peechhe karte rehte ho aur kabhi ek jagah nahi chunte. Do clues madad karte hain: bounded = tum room ke andar rehte ho (ud nahi jaate), monotone = tum hamesha same direction mein chalte ho (sirf aage, ya sirf peechhe). Agar tum sirf aage chalte ho AUR room ke andar rehte ho, toh tumhe kahin na kahin ruk hi jaana padega — yahi Monotone Convergence Theorem hai!

Connections

Completeness of the Real Numbers — woh axiom jo MCT ko power deta hai.
Bolzano–Weierstrass Theorem — bounded ⟹ convergent subsequence.
Series — convergence tests — sequences building blocks hain; partial sums ek sequence hain.
Limits of functions — $ε$ – $δ$ , $ε$ – $N$ ka continuous cousin hai.
Cauchy Sequences — limit ka naam liye bina convergence.
Squeeze Theorem · The number e