Taylor series — derivation from power series

WHAT is a power series?

WHY centre at $a$? Because near $x=a$ the term $(x-a)$ is tiny, so higher powers $(x-a)^n$ shrink fast. The series is most accurate near the centre, and the first few terms dominate (this is the 80/20 of approximation).

HOW to find the coefficients — the derivation

We assume $f$ equals its power series and is differentiable as many times as we want. Our job: solve for each $c_n$.

Step 0 — find $c_0$. Set $x=a$ directly: $f(a) = c_0 + c_1\cdot 0 + c_2 \cdot 0 + \cdots = c_0 \quad\Rightarrow\quad c_0 = f(a)$ Why this step? Every term except the constant has a factor $(x-a)$, which is $0$ at $x=a$.

Step 1 — find $c_1$. Differentiate once: $f'(x) = c_1 + 2c_2(x-a) + 3c_3(x-a)^2 + \cdots$ Now set $x=a$: $\; f'(a) = c_1 \Rightarrow c_1 = f'(a)$. Why this step? Differentiation killed $c_0$ (constant → 0) and shifted $c_1$ into the constant slot.

Step 2 — find $c_2$. Differentiate again: $f''(x) = 2c_2 + (3\cdot 2)c_3(x-a) + (4\cdot3)c_4(x-a)^2 + \cdots$ Set $x=a$: $\; f''(a) = 2c_2 \Rightarrow c_2 = \dfrac{f''(a)}{2}$. Why this step? The power $2$ on $(x-a)^2$ came down as a factor when we differentiated twice: $2\cdot1 = 2!$.

Step $n$ — the pattern. Differentiate $n$ times. The term $c_n(x-a)^n$ becomes $n(n-1)(n-2)\cdots 1 \cdot c_n = n!\,c_n,$ and all lower terms have vanished while higher terms still carry a factor $(x-a)$. Set $x=a$: $f^{(n)}(a) = n!\,c_n \quad\Rightarrow\quad \boxed{c_n = \dfrac{f^{(n)}(a)}{n!}}$

WHY $n!$ appears (the engine of the derivation)

Differentiating $(x-a)^n$ once gives $n(x-a)^{n-1}$, again gives $n(n-1)(x-a)^{n-2}$… Each derivative peels off the current exponent as a multiplying factor. After $n$ derivatives you've multiplied by $n\cdot(n-1)\cdots 2\cdot 1 = n!$. Dividing by $n!$ in the coefficient exactly cancels this, so $c_n$ cleanly equals $f^{(n)}(a)/n!$.

Worked examples

Forecast-then-Verify

Common mistakes

Connections

• Power series — radius & interval of convergence (when does the series actually equal $f$?)
• Taylor's theorem with remainder (the error term $R_n$ that the derivation ignores)
• Maclaurin series — common expansions ($e^x,\sin,\cos,\frac{1}{1-x}$)
• Linear approximation & differentials (the $n=1$ truncation)
• Geometric series (a power series whose coefficients are all $1$)
• L'Hôpital's rule (often replaceable by leading Taylor terms)

Concept Map

Hinglish (regional understanding)