Foundations — Taylor series — derivation from power series

4.3.16 · D1 · Maths › Calculus III — Sequences & Series › Taylor series — derivation from power series

Parent note Taylor series — derivation from power series padhne se pehle, tumhe neeche di gayi das choti ideas khud se samajhni hogi. Hum strictly order mein chalte hain: koi cheez tab tak nahi aayegi jab tak woh define aur draw na ho jaaye.

1. Function $f(x)$ — ek height machine

Picture. $x$ ko horizontal axis par rakho. Output $f(x)$ us point ke upar (ya neeche) ki ek height hai. $x$ ko left-to-right sweep karo aur heights ek curve trace karti hain.

Topic ko yeh kyun chahiye. Taylor series ek curve copy karne ke baare mein hai. Jo cheez hum copy karte hain woh exactly yeh height machine $f(x)$ hai.

2. Point $a$ — woh centre jiske around hum build karte hain

Picture. $x$-axis par $x=a$ par ek single dot mark karo. Jo kuch bhi hum karte hain woh us dot ke paas hota hai.

Kyun. Ek Taylor series sirf ek jagah ke paas accurate hone ka promise karti hai. Woh jagah $a$ hai.

3. Displacement $(x-a)$ — "centre se kitna door"

Picture. $a$ par khade ho jaao. Agar $x$ daayein hai, toh $(x-a)$ ek chota positive step hai; baayein hai toh negative hai; aur $(x-a)=0$ exactly centre par hota hai.

4. Powers aur $x^n$ — repeated multiplication

Kyun. Hamari copy $(x-a{)}^0,(x-a{)}^1,(x-a{)}^2,\dots$ pieces se bani hai — flat piece, sloped piece, curved piece, ... . Har power $n$ ek "shape ingredient" hai.

5. Coefficients $c_n$ — har ingredient ka kitna hissa

Picture. Ek mixing desk socho. $c_0$ base height set karta hai, $c_1$ tilt set karta hai, $c_2$ bend set karta hai. Har dial ghumaana badalta hai ki us ingredient ka kitna hissa jaata hai.

6. Subscript $n$ aur index — terms ginana

Kyun. Hume "general $n$-th term" ke baare mein ek saath baat karni hai, na ki infinitely many lines likhni hain.

7. Sigma notation $\sum$ — compact "sab add karo"

Picture. Ek conveyor belt: counter $n$ clicks $0,1,2,\dots$ aur har click ek term $T_n$ ek running total mein daalta hai.

Kyun. Taylor series literally $\sum _{n=0}^{\infty }{c}_n(x-a{)}^n$ hai. $\sum$ ke bina hum ek infinite sum ek line mein nahi likh sakte.

8. Derivative $f^{\prime }(x)$ — instantaneous steepness

Picture. Curve mein kisi point par tab tak zoom karo jab tak woh seedhi naa lage. $f^{\prime }(x)$ us chote seedhe segment ki steepness hai — tangent line ka rise over run.

9. Higher derivatives $f^{\prime \prime }$, $f^{\prime \prime \prime }$, $f^{(n)}$ — slope ki slope

Kyun. Parent derivation mein, $f^{(n)}(a)$ evaluate karna hi $n$-th coefficient ko pin down karta hai.

10. Factorial $n!$ — multiply-down staircase

Picture. Factors ki ek descending staircase: $n$ se shuru karo aur $1$ tak step down karte jao, har step par multiply karte jao.

11. "$x=a$ par evaluate karna" — freeze trick

Yeh topic ko kaise feed karte hain

Equipment checklist

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

Connections

• Parent: Taylor series — derivation
• Linear approximation & differentials (sirf height + slope match karna)
• Geometric series (simplest power series: saare $c_n=1$)
• Power series — radius & interval of convergence (jab infinite sum allowed bhi ho)
• Maclaurin series — common expansions ($a=0$ wala case jo pehle milta hai)