Term-by-term differentiation and integration of power series

4.3.15 · Maths › Calculus III — Sequences & Series

1. Hum exactly kya claim kar rahe hain?

KYA badalta hai / KYA rehta hai?

Radius $R$ same rehta hai.
Endpoints $x = c \pm R$ par behaviour badal sakta hai (integration se convergence aa sakti hai, differentiation se ja sakti hai). Endpoints ko hamesha alag se check karo.

2. Radius kyun nahi badlta? (Derivation scratch se)

Hum ise chhod nahi dete — hum ise kamate hain.

Derivation. Radius (Cauchy–Hadamard ke zariye) is cheez se govern hota hai: $\frac{1}{R} = lim sup_{n \to \infty} ∣ a_{n} ∣^{1/ n} .$

Differentiated series ke coefficients $b_{n} = (n + 1) a_{n + 1}$ hain (re-indexing ke baad). Uska radius $R^{'}$ satisfy karta hai: $\frac{1}{R ^{'}} = lim sup_{n \to \infty} ∣ (n + 1) a_{n + 1} ∣^{1/ n} .$

Ab ek crucial limit use karo, jise hum prove karte hain kyunki yahi puri engine hai: $lim_{n \to \infty} n^{1/ n} = 1.$

Isliye $∣ (n + 1) ∣^{1/ n} \to 1$ aur $lim sup$ ki value unchanged rehti hai: $lim sup ∣ (n + 1) a_{n + 1} ∣^{1/ n} = lim sup ∣ a_{n + 1} ∣^{1/ n} = \frac{1}{R} .$ Toh $R^{'} = R$ . Integration ka case identical hai factor $\frac{1}{n + 1}$ ke saath, jiska $1/ n$ -th power bhi $\to 1$ hota hai. ∎

3. Hum $\frac{d}{d x}$ aur $\sum$ ko swap kar hi kyun sakte hain?

4. Worked Examples

Figure — Term-by-term differentiation and integration of power series

5. Forecast-then-Verify drill

Recall Forecast:

$\sum_{n = 0}^{\infty} \frac{x ^{n}}{n !}$ ko differentiate karo. Tumhara prediction kya hai? Prediction: ise khud wapas dena chahiye ( $e^{x}$ ). Verify: $\frac{d}{d x} \sum \frac{x ^{n}}{n !} = \sum_{n = 1}^{\infty} \frac{n x ^{n - 1}}{n !} = \sum_{n = 1}^{\infty} \frac{x ^{n - 1}}{( n - 1 )!} = \sum_{m = 0}^{\infty} \frac{x ^{m}}{m !} .$ ✓ Series apni khud ki derivative ke barabar hai — $e^{x}$ ke saath consistent.

6. Flashcards

Jab aap ek power series ko term by term differentiate karte ho toh radius of convergence ka kya hota hai?

Woh same rehta hai (

R

unchanged); sirf endpoint behaviour badal sakta hai.

n^{1/ n} \to 1

kyun hota hai?

Kyunki

ln (n^{1/ n}) = \frac{l n n}{n} \to 0

, toh

n^{1/ n} = e^{l n n / n} \to e^{0} = 1

\sum a_{n} (x - c)^{n}

ka term-by-term derivative kya hai?

\sum_{n = 1}^{\infty} n a_{n} (x - c)^{n - 1}

\sum a_{n} (x - c)^{n}

ka term-by-term integral kya hai?

C + \sum_{n = 0}^{\infty} \frac{a _{n}}{n + 1} (x - c)^{n + 1}

Integration kabhi kya gain kar sakti hai jo differentiation lose kar sakta hai?

Ek endpoint

x = c \pm R

par convergence.

Power series mein

\frac{d}{d x}

aur

\sum

ko swap karne ki analytic property kya justify karti hai?

(c - R, c + R)

ke compact subintervals par derivative series ki uniform convergence.

arctan x

ki series kaise derive hui?

\frac{1}{1 + x ^{2}} = \sum (- 1)^{n} x^{2 n}

ko term by term integrate karo:

\sum \frac{( - 1 ) ^{n}}{2 n + 1} x^{2 n + 1}

π /4

ka Leibniz formula kahan se aata hai?

arctan

series mein

x = 1

rakho:

1 - \frac{1}{3} + \frac{1}{5} - \dots = \frac{π}{4}

Recall Feynman: ek 12-saal ke bachche ko explain karo

Ek bahut lamba polynomial socho jo kabhi khatam nahi hota. Normal chote polynomials aasaan hote hain: slope nikalne ke liye sirf power neeche laao, aur area nikalne ke liye power upar bump karo. Cool baat yeh hai ki yeh kabhi na khatam hone wala wala bhi usi simple rule se chalta hai jab tak tum uski "safe zone" ke andar ho (yani interval ke andar). Aur safe zone ka size bilkul same rehta hai is karne ke baad — sirf do endpoints (kinare) shayad behave karna shuru ya band kar dein. Toh agar tum ek infinite-polynomial jaante ho, toh har piece ko slope ya area karke free mein naye wale bana sakte ho.

Connections

Power series and radius of convergence
Cauchy–Hadamard theorem
Uniform convergence and the Weierstrass M-test
Taylor and Maclaurin series
Geometric series
Abel's theorem (endpoint behaviour)
Solving ODEs with power series