Foundations — Power series — centre, radius of convergence, interval of convergence

4.3.14 · D1 · Maths › Calculus III — Sequences & Series › Power series — centre, radius of convergence, interval of co

Radius, interval, ya Cauchy–Hadamard formula se milne se pehle, tumhe har woh piece of notation apni ho leni chahiye jo parent note use karta hai. Hum unhe order mein build karte hain — ladder ki har rung ke liye uske neeche wali zaruri hai.

1. Summation symbol $\sum$

Picture yeh hai: ek row of boxes socho, har $n$ ki value ke liye ek. Box $n$ mein number $t_n$ hai. $\sum$ bas kehta hai "har box ko ek running total mein daalo."

Topic ko yeh kyun chahiye: ek power series hai hi ek $\sum$ jisme infinitely many boxes hain. Is symbol ke bina hum woh object likh bhi nahi sakte jo hum padh rahe hain.

2. Infinity $\infty$ aur "converges" ka matlab

Picture yeh hai: number line par dots march karte hue. Converging = dots ek spot par zyada se zyada tight hote jaate hain. Diverging = ya to woh hamesha ke liye right ki taraf nikal jaate hain ya bina kahin land kiye bounce karte rehte hain.

Topic ko yeh kyun chahiye: ek infinite sum ka value tab hi hota hai jab woh converge kare. Poora khel — radius aur interval dhundhna — exactly yeh dhundhna hai ki kaunse inputs ise converge karaate hain.

3. Absolute value $|\cdot |$ — distance, sign nahi

Picture yeh hai: $a$ par ek pin lagao. Tab $|x-a|$ woh gap hai — segment ki length — tumhare test point $x$ se pin tak, chahe $x$ kisi bhi taraf ho.

Topic ko yeh kyun chahiye: convergence centre $a$ se symmetrically failti hai. "Main kitna door ja sakta hoon?" ek distance ka sawaal hai, isliye condition $|x-a|<R$ likhi jaati hai. Yeh single distance inequality hi wajah hai ki jawab hamesha ek interval hota hai — $a$ ke dono directions equal treat hoti hain.

4. Centre $a$, coefficients $c_n$, aur power series khud

Picture yeh hai: ek ordinary polynomial jaise $3+2y+5y^2$, lekin $y=x-a$ ke saath (pin se measure karte hue) aur coefficients ki list hamesha ke liye chalti rehti hai.

Topic ko yeh kyun chahiye: yeh topic ke chaar nouns hain. $a$ aur $c_n$ tumhe diye jaate hain; $x$ woh hai jo tum test karte ho; powers $(x-a{)}^n$ hi woh cheez hai jo door ke $x$ ko dangerous banati hai — ek bada $(x-a)$ jo high power par uthaya jaaye woh explode kar sakta hai.

5. $n$-th root $\sqrt[n]{}$ aur exponent $n$

Topic ko yeh kyun chahiye: master formula $\sqrt[n]{|c_n|}$ use karta hai. Ek term ka size dekho, $|c_n||x-a{|}^n$. $n$-th root lene par $|x-a{|}^n$ wapas clean $|x-a|$ ban jaata hai: $\sqrt[n]{|c_n||x-a{|}^n}=\sqrt[n]{|c_n|}\cdot |x-a|.$ Yahi poora trick hai — root input se power hata deta hai taaki $x$ wala part factor out ho sake. Root Test dekho.

6. Limit $\lim$ — woh value jo approach ki ja rahi hai

Picture yeh hai: $\frac{n}{n+1}$ deta hai $\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\dots$ — dots hamesha ke liye $1$ ki taraf creep karte hain. Limit $1$ hai.

Topic ko yeh kyun chahiye: dono convergence tests ek limiting ratio ya root measure karte hain. "Kya yeh converge karta hai?" yeh decide hota hai ki terms ultimately kahan ja rahi hain, kisi bhi ek early term se nahi.

7. Limit superior $\mathrm{limsup}$ — honest ceiling jab koi single limit nahi

Picture yeh hai: ek sequence jo ek low band aur ek high band ke beech hop karti hai. Sabse chhoti horizontal ceiling banao jisko dots baar baar poke karte rehte hain. Woh ceiling ki height $\mathrm{limsup}$ hai.

Topic ko yeh kyun chahiye: master (Cauchy–Hadamard) formula hai $1/R={\mathrm{limsup}}_n\sqrt[n]{|c_n|}$. Coefficients oscillate kar sakte hain (parent mein Example 5), isliye plain $\lim$ fail ho sakta hai — lekin $\mathrm{limsup}$ hamesha exist karta hai $[0,\infty ]$ mein. Yahi wajah hai ki guaranteed tool woh hai, $\lim$ nahi. Poori detail Limit Superior and Limit Inferior mein hai.

8. Comparison inequalities $<,>,\le ,\ge$ aur $R$

Topic ko yeh kyun chahiye: $R$ convergence ki reach hai jo pin $a$ se distance ke roop mein measure hoti hai. Strict $<$ / $>$ cleanly split hote hain, lekin $=$ case hi woh poori wajah hai ki hum endpoint tests alag se karte hain — Ratio Test, Alternating Series Test, p-series dekho.

Prerequisite map

Equipment checklist

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

Taiyaar? parent topic par wapas jao aur derivation plain English jaisi lagegi.