Sequences — convergence, divergence, boundedness, monotonicity

WHAT is a sequence?

The first 5, 50, or 5 million terms NEVER decide convergence — only the tail does.

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

We want to say " $a_n$ gets close to $L$ and stays close." How do we make "close" precise without hand-waving?

WHY the $\varepsilon$ – $N$ definition? Because "gets close" is vague. So we let a challenger pick any tolerance $\varepsilon>0$ (a target band around $L$ ). We must respond with a cutoff $N$ so that every term past $N$ lies inside that band. If we can always respond, no matter how tiny $\varepsilon$ , the terms truly home in on $L$ .

Divergence — the flavours

Boundedness — a necessary condition

Monotonicity — the second tool

HOW to test: compute $a_{n+1}-a_n$ (sign?) or the ratio $a_{n+1}/a_n$ vs $1$ (for positive terms), or differentiate $f(x)$ where $a_n=f(n)$ .

The payoff: Monotone Convergence Theorem (MCT)

Figure — Sequences — convergence, divergence, boundedness, monotonicity

Worked example Recursive:

$a_1=2,\ a_{n+1}=\tfrac12\!\left(a_n+\tfrac{2}{a_n}\right)$

Bounded below: by AM–GM, $a_{n+1}=\tfrac12(a_n+\tfrac2{a_n})\ge\sqrt{2}$ . (Why AM–GM? it gives $\tfrac12(x+y)\ge\sqrt{xy}$ with $xy=2$ .)
Decreasing: $a_n-a_{n+1}=a_n-\tfrac12(a_n+\tfrac2{a_n})=\tfrac{a_n^2-2}{2a_n}\ge0$ since $a_n\ge\sqrt2$ . (Why? numerator $\ge0$ once bounded below.)
Decreasing + bounded below ⟹ converges to $L$ . Solve $L=\tfrac12(L+\tfrac2L)\Rightarrow L^2=2\Rightarrow L=\sqrt2$ . (Why this works: the limit satisfies the same fixed-point equation.)

Useful limit tools (so you don't always use $\varepsilon$ – $N$ )

Forecast-then-Verify drill

Common mistakes (Steel-manned)

Flashcards

What is a sequence, formally?

A function

a:\mathbb{N}\to\mathbb{R}

, written

(a_n)

State the

\varepsilon

–

N

definition of

a_n\to L

\forall\varepsilon>0\,\exists N: n>N\Rightarrow|a_n-L|<\varepsilon

Convergent ⟹ ?

Bounded (proof:

\varepsilon=1

tail + finite head).

Is bounded ⟹ convergent? Give counterexample.

No;

a_n=(-1)^n

is bounded but oscillates.

State the Monotone Convergence Theorem.

A monotone bounded sequence converges (to its sup/inf).

Which axiom underlies MCT?

Completeness of

\mathbb{R}

(least upper bound property).

How do you find the limit of a convergent recursive sequence

a_{n+1}=f(a_n)

Solve the fixed point

L=f(L)

State the Squeeze Theorem for sequences.

b_n\le a_n\le c_n

and

b_n,c_n\to L

then

a_n\to L

Why does

(-1)^n

diverge (one line)?

Tail can't fit in a width-

<2

band while terms stay distance

2

apart.

Does changing the first 1000 terms change convergence?

No — only the tail matters.

Test for monotonicity of

(a_n)

Sign of

a_{n+1}-a_n

, or

a_{n+1}/a_n

1

for positive terms.

Limit of

(1+1/n)^n

e

(increasing + bounded above by 3 ⟹ MCT).

Recall Feynman: explain to a 12-year-old

Imagine you're walking toward a wall and each step you take is half the leftover distance. Your footprints make a list of positions: that list is a sequence. Converging means you keep getting closer and closer to the wall and basically end up touching it. Diverging means you either run away forever, or you keep hopping back and forth and never pick a spot. Two clues help: bounded = you stay inside the room (don't fly off), monotone = you always move the same direction (only forward, or only back). If you only move forward AND you stay in the room, you must end up stopping somewhere — that's the Monotone Convergence Theorem!

Connections

Completeness of the Real Numbers — the axiom powering MCT.
Bolzano–Weierstrass Theorem — bounded ⟹ convergent subsequence.
Series — convergence tests — sequences are the building blocks; partial sums are a sequence.
Limits of functions — $\varepsilon$ – $\delta$ is the continuous cousin of $\varepsilon$ – $N$ .
Cauchy Sequences — convergence without naming the limit.
Squeeze Theorem · The number e

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek sequence matlab numbers ki ek infinite list — jaise $a_1, a_2, a_3,\dots$ — jo asal mein ek function hai $\mathbb{N}\to\mathbb{R}$ . Sabse bada sawaal sirf ek hota hai: jab $n$ bahut bada hota jaata hai, kya terms kisi ek number ke paas jaake settle ho jaate hain? Agar haan, to converge; agar nahi (ya to bhaag jaate hain infinity ki taraf, ya bounce karte rehte hain), to diverge.

"Paas jaana" ko exact banane ke liye $\varepsilon$ – $N$ definition use karte hain: koi bhi chhota band $\varepsilon$ tum chuno limit $L$ ke aaspaas, mujhe ek cutoff $N$ dena padega jiske baad saare terms us band ke andar aa jaayein. Yeh "challenge–response" game hi convergence ka asli matlab hai. Yaad rakho — sirf tail (aage ke terms) matter karta hai, pehle ke 1000 terms badal do, koi farak nahi padta.

Do tools bahut kaam aate hain. Bounded matlab sequence kamre ke andar rehti hai, bhaagti nahi. Monotone matlab hamesha ek hi direction — ya sirf badhti, ya sirf ghatti. Yahan ek important point: sirf bounded hona kaafi NAHI hai (jaise $(-1)^n$ bounded hai par bounce karta hai). Lekin Monotone + Bounded = Converge — yeh hai MCT (Monotone Convergence Theorem), jo $\mathbb{R}$ ki completeness se aata hai.

Recursive sequences mein ek trick: agar pata chal gaya ki converge karti hai, to limit nikaalne ke liye fixed point equation $L=f(L)$ solve karo — jaise Newton ki $\sqrt2$ wali sequence mein $L=\tfrac12(L+\tfrac2L)$ se $L=\sqrt2$ aata hai. Bas yeh samajh lo aur Squeeze Theorem ( $\frac{\sin n}{n}\to 0$ type) — pura topic clear ho jaayega.

Sequences — convergence, divergence, boundedness, monotonicity

WHAT is a sequence?

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

Divergence — the flavours

Boundedness — a necessary condition

Monotonicity — the second tool

The payoff: Monotone Convergence Theorem (MCT)

Useful limit tools (so you don't always use $\varepsilon$ – $N$ )

Forecast-then-Verify drill

Common mistakes (Steel-manned)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Calculus III — Sequences & Series

Connections