Ratio test — proof, limitations: Foundations — Hinglish deep dive

4.3.12 · D1 · Maths › Calculus III — Sequences & Series › Ratio test — proof, limitations

Hum parent note Ratio Test — Proof & Limitations ko padhne se pehle, usmein likhe har squiggle ko samajhna zaroori hai. Neeche, har symbol ko teen cheezein milti hain: plain words, ek picture, aur topic ko yeh kyun chahiye. Inhe is tarah order kiya gaya hai ki har ek sirf woh ideas use karta hai jo pehle se build ho chuke hain.

1. Ek sequence: $a_{n}$

Picture: numbered mailboxes ki ek row. Box number $n$ mein number $a_{n}$ rakha hai.

Figure — Ratio test — proof, limitations

Topic ko yeh kyun chahiye: ratio test terms ki ek list ke baare mein hai aur ek term ka uske neighbour se kya relation hai. Koi sequence nahi, toh test karne ko kuch bhi nahi.

2. Neighbour: $a_{n + 1}$

Picture: box $n$ par khado aur ek darwaza daayein dekho. Woh neighbour hai $a_{n + 1}$ .

Topic ko yeh kyun chahiye: poora test har term ko uske theek baad waale term se compare karta hai. "Agla term" ka naam liye bina, hum ratio bhi nahi likh sakte.

3. Sab add karna: $\sum a_{n}$

Picture: mailbox values ko ek doosre ke upar stack karo ek badhte tower mein jis ki height pehle $a_{1}$ , phir $a_{1} + a_{2}$ , phir $a_{1} + a_{2} + a_{3}$ hoti hai, …

Topic ko yeh kyun chahiye: poora sawaal — kya yeh converge karta hai? — is tower ke baare mein hai. Kya yeh ek finite height par settle ho jaata hai, ya infinity tak shoot karta hai?

4. Partial sums aur "convergence"

Picture: jaise tum blocks add karte ho, tower ki top edge dekho. Converge = edge ek horizontal line ki taraf ghanti hai. Diverge = edge hamesha upar ki taraf jaati rehti hai.

Topic ko yeh kyun chahiye: "converges" aur "diverges" woh do verdicts hain jo ratio test deta hai. Yahi in shabdon ka matlab hai.

5. Absolute value: $∣ x ∣$

Picture: negative number line ko positive side par fold karo; $∣ x ∣$ hai ki tum origin se kitni door land karte ho.

6. Ratio: $\frac{a _{n + 1}}{a _{n}}$

Picture: do blocks side by side. Naye block ki height ko puraane ek ke ek multiple ke roop mein measure karo.

Topic ko yeh kyun chahiye: yeh akela number test ka dil hai. Ratio kyun aur difference kyun nahi? Kyunki hamara benchmark, geometric series, repeated multiplying se bana hai — isliye natural comparison yeh hai ki "humne kis se multiply kiya," jo exactly ek ratio hai.

7. Benchmark: geometric series $\sum r^{k}$

Picture: har block pehle waale ka fixed fraction (maano har ek pehle waale ka $\frac{2}{3}$ ). Tower visibly ek ceiling ki taraf close in karta hai.

Topic ko yeh kyun chahiye: yeh akela series hai jiska fate hum inspection se definitely jaante hain. Ratio test tumhari unknown series ko is known series ke against squeeze karke kaam karta hai — woh squeeze hai Comparison Test.

8. Limit: $L = n \to \infty lim \frac{a _{n + 1}}{a _{n}}$

Picture: shrink-ratios $\frac{a _{2}}{a _{1}}, \frac{a _{3}}{a _{2}}, \dots$ ko dots ki tarah plot karo. $L$ woh horizontal line hai jise woh long run mein hug karte hain.

Topic ko yeh kyun chahiye: $L$ test ka output hai. Teeno verdicts — $L < 1$ converge, $L > 1$ diverge, $L = 1$ inconclusive — sab is ek number ke baare mein statements hain.

9. Infinity aur " $\infty$ "

Picture: ratios ka dot-plot page ke top se upar jaata hua, kabhi level nahi hota.

Prerequisite map

Ise top to bottom padho: mailboxes ( $a_{n}$ ) aur unke neighbours ( $a_{n + 1}$ ) ek ratio dete hain; absolute value mein wrap karne se yeh ek size ban jaata hai; limit poori tail ko ek number $L$ mein badal deta hai; saath mein geometric series ek known benchmark deti hai, aur uske against comparison squeeze verdict deliver karta hai — yahi ratio test hai.

Equipment checklist

Khud test karo — jawaab dene ke baad hi reveal karo.

Index

n

kya represent karta hai, aur

a_{n}

kya hold karta hai?

n

ek counting-number address hai (

1, 2, 3, \dots

);

a_{n}

woh value hai jo us address par stored hai.

a_{n + 1}

words mein kya hai?

Bilkul agla term,

a_{n}

ke ek step baad.

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

kya instruction deta hai?

n = 1

se shuru karo aur har term hamesha add karte raho.

Partial sum

S_{N}

kya hai?

Sirf pehle

N

terms ka running total:

a_{1} + \dots + a_{N}

Ek series converge kab karti hai?

Jab uske partial sums

S_{N}

N \to \infty

hone par ek finite number par settle ho jaayein.

∣ x ∣

geometrically kya hai?

x

0

se distance — uska size, sign ignore karke.

Test ratio ko

∣ \cdot ∣

mein kyun wrap karta hai?

Yeh term sizes compare karta hai; signs geometric comparison ko break kar dete, isliye hum unhe strip karte hain.

Shrink-ratio

\frac{a _{n + 1}}{a _{n}}

kya measure karta hai?

Woh factor jis se ek term ka size pehle waale term ke comparison mein change hota hai.

Ek geometric series ka common ratio

r

kya hai?

Woh fixed number jis se har term ko multiply karke agla term milta hai.

\sum r^{k}

kab converge karta hai?

Exactly jab

∣ r ∣ < 1

L = lim_{n \to \infty} \frac{a _{n + 1}}{a _{n}}

kya report karta hai?

Shrink-ratio ki long-run, settled value (tail ka behaviour).

L = \infty

ka kya matlab hai?

Ratios bina bound ke badhte hain; terms explode ho jaate hain aur series diverge karti hai.

Connections

Geometric Series — woh benchmark jis ki taraf yahan har symbol build ho raha hai.
Comparison Test — woh squeeze jo benchmark ko verdict mein badalta hai.
Term Test (nth-term divergence) — $L > 1$ case ke peeche quick $a_{n} \to 0$ check.
Back to the parent: Ratio Test