Foundations — Convergence tests for improper integrals — comparison

4.2.12 · D1 · Maths › Calculus II — Integration › Convergence tests for improper integrals — comparison

Yeh page maanta hai ki aapne pehle kuch nahi dekha. Isse pehle ki aap judge kar sako ki ${\int }_1^{\infty }\frac{dx}{x^2+5}$ converge karta hai ya nahi, aapko page par har ek nishan — $\int$, $\infty$, $\le$, aur "limit" shabd — apna banana hoga. Hum har cheez ek picture se build karte hain, uss order mein jis order mein woh ek doosre par depend karti hain.

1. Curve ke neeche ka area — $\int$ asal mein kya draw karta hai

Picture. Region ko patli vertical strips mein kaat lo. Har strip kisi $x$ par baith'ti hai, uski height $f(x)$ hai, aur ek choti si width hai jise hum $dx$ kehte hain. Uska area hai height $\times$ width $=f(x)dx$. Stretched-S symbol $\int$ purana letter S jiska matlab hai "Sum" — iska matlab hai "un saari patli strips ko jod do." $a$ aur $b$ limits of integration hain: jahan summing shuru hoti hai aur khatam hoti hai.

Yeh topic ko kyun chahiye. Comparison mein har claim ("yeh area finite / infinite hai") ek claim hai $\int$ ke baare mein. Agar $\int$ aapke dimag mein accumulated area ki picture nahi hai, toh koi bhi test kuch matlab nahi rakhta.

2. Symbols $f$, $x$, $g$ — functions height-machines ki tarah

Picture. Floor par position $x$ par khade ho jao. Seedha upar dekho jab tak curve se na takra lo. Jo height aapne chadhee woh $f(x)$ hai. Is poore topic mein $f$ woh mushkil function hai jiska area hum compute nahi kar sakte, aur $g$ woh aasaan benchmark function hai jisse hum compare karte hain. Woh ek hi axes par rehte hain, toh hum literally dekh sakte hain ki kaun sa curve upar hai.

Yeh topic ko kyun chahiye. "$f$ ko $g$ se compare karo" ka matlab hai "har position $x$ par, kya $f$ ki height $g$ ki height se neeche hai ya upar?" Yeh ek baat hai jo aapko dikhni chahiye.

3. Inequality $0\le f(x)\le g(x)$ — ek curve doosre ke neeche trap

$0\le f$ kyun bhi? Extra "$0\le$" kehta hai ki $f$ kabhi floor ke neeche nahi jaati — uska area sirf badhta hai jab hum window bada karte hain, kabhi ghattha nahi. Figure dekho: blue $f$ floor se chipki hui hai, orange $g$ uske upar float kar rahi hai. Kyunki blue floor aur orange ke beech strip mein squeeze hai, blue area kabhi bhi orange area se zyada nahi ho sakta. Yeh akela sentence Direct Comparison Test ka poora engine hai — pardon, parent ke Direct Comparison Test ka.

Yeh topic ko kyun chahiye. DCT literally yahi inequality plus ek logical step hai. $\le$ ki direction galat kar do aur har conclusion palat jaata hai.

4. Symbol $\infty$ aur kyun hum limit se iske paas jaate hain

Picture. Maano $G(t)={\int }_1^{t}fdx$ $1$ se sliding wall $t$ tak ka area hai. Jab aap deewar ko daayein khinchte ho, zyada area accumulate hota hai, toh $G(t)$ chadh'ta hai. Do cheezein ho sakti hain:

• Yeh level off ho jaata hai ek ceiling ki taraf — total area ek finite number hai. Hum kehte hain improper integral converges.
• Yeh bina ceiling ke badhta rehta hai — total area infinite hai. Yeh diverges.

Yeh topic ko kyun chahiye. "Converges" aur "diverges" is limit se define hote hain. Comparison ka poora point yeh decide karna hai ki figure ki do red curves mein se aapka unknown area kaun si curve follow karta hai — bina curve exactly compute kiye.

5. Powers $x^{-p}$ aur benchmark idea

Picture. $\frac{1}{x}$ (dheere gir'ta hai) ko $\frac{1}{x^2}$ (tezi se gir'ta hai) se compare karo. Jo teji se gir'ta hai woh apni lambi tail mein kam area chodh'ta hai. Ek knife-edge exponent hai, $p=1$, jo "tail area finite" aur "tail area infinite" ko divide karta hai:

${\int }_1^{\infty }\frac{dx}{x^p}\text{converges}⟺p>1.$

Yeh topic ko kyun chahiye. Lagbhag har benchmark $g$ ek power $x^{-p}$ hai. Akela fact "$p>1$ ⇒ finite, $p\le 1$ ⇒ infinite" woh yardstick hai jiske against har comparison measure karta hai. Full derivation ke liye The p-integral and p-series dekho jo parent deta hai.

6. Ratio $L=\lim f/g$ — heights ki jagah growth rates compare karna

Picture. Socho $f$ aur $g$ daayein march kar rahe hain. Agar unki heights ek fixed proportion mein rehti hain (maano $f$ hamesha roughly $3$ times $g$ hai), toh jahan bhi ek ka total area finite hai, doosre ka bhi hai; jahan bhi ek infinite hai, doosre ka bhi. Ratio $L$ us stable proportion ko measure karta hai.

Yeh topic ko kyun chahiye. Jab inequality $f\le g$ haath se prove karna mushkil ho, Limit Comparison Test aapko laraai skip karne deta hai: sirf check karo ki dominant terms match karti hain. Yeh parent ke Example 2 ke peeche wala tool hai, jahan $\frac{1}{\sqrt{x}+x}$ $\frac{1}{x}$ jaisa behave karta hai.

Yeh foundations topic ko kaise feed karte hain

Equipment checklist

Khud test karo — daayein side cover karo aur zor se jawab do.

Connections

• Yeh note Hinglish mein padho →
• The p-integral and p-series
• Improper integrals — infinite discontinuities (type 2)
• Monotone Convergence Theorem
• Comparison test for infinite series
• Integral Test for series
• Absolute vs conditional convergence