Ratio test — proof, limitations

4.3.12 · Maths › Calculus III — Sequences & Series

WHAT it says

Quantity $|a_{n+1}/a_n|$ local shrink-ratio hai. $L$ uski long-run value hai.

WHY it works — derivation from scratch

Proof ek geometric series ke saath comparison hai. Hum khud isko build karte hain.

Case $L<1$ (convergence).

Yeh step kyun? Kyunki $L<1$ hai, $L$ aur $1$ ke beech "jagah" hai. Us gap mein ek number $r$ choose karo.

Choose $r$ with $L<r<1$. Limit ki definition se, kyunki $|a_{n+1}/a_n|\to L<r$, eventually ratio $r$ ke neeche rehti hai: $\exists N\text{ such that }n\ge N⟹\left|\frac{a_{n+1}}{a_n}\right|<r.$

Yeh step kyun? Yeh limit ko unpack karna hi hai: ek baar $N$ ke baad, har shrink-ratio $r$ ke under hai.

Ab $N$ ke baad se inequalities chain karo: $|a_{N+1}|<r|a_N|,|a_{N+2}|<r|a_{N+1}|<r^2|a_N|,\dots$ Induction se: $|a_{N+k}|<r^k|a_N|.$

Yeh step kyun? Har term zyada se zyada apne pehle wale ka $r$ times hai, $k$ baar apply karne par $r^k$ milta hai.

To tail term-by-term ek geometric series se bounded hai: ${\sum }_{k=0}^{\infty }|a_{N+k}|<|a_N|{\sum }_{k=0}^{\infty }{r}^k=\frac{|a_N|}{1-r}<\infty .$

Yeh step kyun? $\sum r^k$ converge karta hai kyunki $0<r<1$ hai. Ek aisi series jiske terms ek convergent positive series se dominated hain, converge karti hai (comparison test). $N$ se pehle ke finitely many terms add karne se yeh finite rehta hai. Isliye $\sum a_n$ absolutely converge karta hai. ∎

Case $L>1$ (divergence).

Choose $r$ with $1<r<L$. Tab eventually $\left|\frac{a_{n+1}}{a_n}\right|>r>1⟹|a_{n+1}|>|a_n|.$

Yeh step kyun? $N$ ke baad terms grow kar rahi hain, isliye $|a_n|\nrightarrow 0$. Lekin kisi bhi series ke converge hone ke liye ek zaroori condition hai $a_n\to 0$ (term test). Jab yeh fail hoti hai, series diverge karti hai. ∎

WHY $L=1$ fails — the limitation

Baaki limitations:

• $a_n\ne 0$ (ya eventually nonzero) chahiye.
• Yeh absolute convergence ke baare mein batata hai; conditional convergence ke liye (jaise alternating harmonic) jab $L=1$ ho, alternating series test ki zaroorat hoti hai.
• Best tab hota hai jab terms mein factorials ya exponentials ($n!$, $r^n$) hon, jahan ratios cleanly simplify hote hain.

Worked examples

Flashcards

Connections

• Geometric Series — woh foundation jis par proof tikha hai.
• Comparison Test — woh mechanism jo convergence case khatam karta hai.
• Term Test (nth-term divergence) — jo $L>1$ case ko khatam karta hai.
• Root Test — sibling test, aksar tab kaam karta hai jab ratio $L=1$ de.
• p-Series and Integral Test — sahi tool jab ratio test fail ho ($L=1$).
• Radius of Convergence — power series par direct application.
• Raabe's Test — $L=1$ borderline ke liye refinement.

Concept Map