Direct comparison test — Hinglish notes

4.3.8 · Maths › Calculus III — Sequences & Series

YEH HAI KYA?

YEH KAAM KYUN KARTA HAI? (Derivation scratch se)

Hum ise Monotone Convergence Theorem se banate hain: ek sequence jo increasing aur upar se bounded ho, converge karti hai.

Maano $a_{n} \geq 0$ . Partial sums define karo: $S_{n} = a_{1} + a_{2} + \dots + a_{n} .$

Step 1 — Partial sums increasing hain. $S_{n + 1} - S_{n} = a_{n + 1} \geq 0$ , toh $S_{n}$ monotone increasing hai. Yeh step kyun? Terms ki non-negativity hi guarantee karti hai ki partial sums kabhi neeche nahi jaate — isliye DCT ko $a_{n} \geq 0$ chahiye.

Step 2 — Partial sums ko bound karo. Kyunki $a_{n} \leq b_{n}$ hai, $S_{n} = \sum_{k = 1}^{n} a_{k} \leq \sum_{k = 1}^{n} b_{k} \leq \sum_{k = 1}^{\infty} b_{k} =: B .$ Yeh step kyun? Agar $\sum b_{n}$ converge kare, toh uske partial sums finite total $B$ se bounded hain. Term-by-term domination woh bound $S_{n}$ tak pass kar deti hai.

Step 3 — Monotone Convergence lagao. $S_{n}$ increasing aur $B$ se upar bounded hai. Isliye $S_{n}$ ek finite limit par converge karta hai. Hence $\sum a_{n}$ converge karta hai. ∎

Contrapositive se divergence wala half milta hai. Agar $\sum a_{n}$ diverge kare, toh $S_{n} \to \infty$ . Kyunki $\sum_{k = 1}^{n} b_{k} \geq S_{n}$ , bade partial sums bhi $\to \infty$ hain, toh $\sum b_{n}$ diverge karta hai. ∎

ISKO USE KAISE KARTE HAIN (ek recipe)

Ensure karo ki terms eventually $\geq 0$ hain.
Convergence ya divergence guess karo (dominant power dekho → $p$ -series / geometric socho).
Inequality sahi direction mein banao:
- Convergence prove karne ke liye, ek bada convergent $b_{n}$ dhundho.
- Divergence prove karne ke liye, ek chota divergent $b_{n}$ dhundho.
Inequality ko algebraically justify karo.
Known benchmark result batao aur conclude karo.

Worked Examples

Common Mistakes (Unhe Samjho)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum hamesha ke liye chhoti-chhoti candy ki dher jod rahe ho. Sab khud se nahi jod sakte, toh tum cheat karte ho: tum ek alag candy ki dher dhundho jise tum pehle se samajhte ho.

Agar tumhari dher hamesha choti ho ek aisi se jo ek finite jar mein add ho jaati hai, toh tumhari dher bhi ek jar mein fit ho jaayegi — woh badhna band ho jaati hai. (converges)
Agar tumhari dher hamesha badi ho ek aisi se jo infinity tak grow karti hai, toh tumhari dher bhi infinity tak grow karegi. (diverges) Tum apni khud ki candies kabhi count nahi karte — bas pile-to-pile kisi trusted dost se compare karte ho!

Flashcards

Direct Comparison Test terms par kya do conditions chahiye?

Woh non-negative honi chahiye (

0 \leq a_{n} \leq b_{n}

), kam se kam eventually (for

n \geq N

Agar

0 \leq a_{n} \leq b_{n}

aur

\sum b_{n}

converge kare, toh

\sum a_{n}

ka kya hoga?

Woh converge karega (finite roof ke neeche trap hai).

Agar

0 \leq a_{n} \leq b_{n}

aur

\sum a_{n}

diverge kare, toh

\sum b_{n}

ka kya hoga?

Woh diverge karega (infinite floor ke upar push hai).

DCT ki proof ke peeche kaunsa theorem hai?

Monotone Convergence Theorem (increasing + bounded above ⇒ converges).

DCT ke liye terms non-negative kyun honi chahiye?

Taaki partial sums monotone increasing rahen, jisse Monotone Convergence apply ho sake.

DCT se ek series ko CONVERGE prove karne ke liye, tum use ek ___ series se compare karte ho jo ___ ho.

bigger; convergent.

DCT se ek series ko DIVERGE prove karne ke liye, tum use ek ___ series se compare karte ho jo ___ ho.

smaller; divergent.

Kya

a_{n} \leq \frac{1}{n ^{2}}

dikhana (smaller, convergent) prove karta hai ki

\sum a_{n}

converge karta hai?

Nahi — woh useless direction hai;

a_{n}

phir bhi diverge ho sakta hai.

\sum \frac{2 + s i n n}{n ^{2}}

ke liye, kaunsa bound convergence deta hai?

\frac{2 + s i n n}{n ^{2}} \leq \frac{3}{n ^{2}}

, aur

\sum 3/ n^{2}

converge karta hai.

Jab terms ka sign badle toh DCT ki jagah kaunsa test use karte hain?

Absolute values compare karo (absolute convergence) ya Alternating Series Test use karo.

Connections

p-series test — sabse common benchmark ( $p > 1$ converge karte hain, $p \leq 1$ diverge).
Geometric series — ek aur common comparison partner.
Limit comparison test — DCT ka gentle cousin jab inequality awkward ho.
Monotone convergence theorem — proof ke peeche ka engine.
Harmonic series — canonical divergent benchmark.
Absolute convergence — jab terms sab positive nahi hote tab kya karo.
Integral test — benchmark series classify karne ka alternate tarika.