Chapter 6 — Taylor & Maclaurin Series

By the end of this chapter you will:

1. Derive the Taylor coefficient formula $c_n = \frac{f^{(n)}(a)}{n!}$
2. Write the Taylor / Maclaurin series of a function
3. Use Taylor polynomials (partial sums) for numerical estimation
4. Understand why the coefficients take exactly that form

6.1 Motivation — Approximating Functions with Polynomials

Problem: can a polynomial $P(x) = c_0 + c_1 x + c_2 x^2 + \cdots$ approximate a complicated function $f(x)$ — and grow more accurate with more terms?

Why we care: polynomials are easy to compute (just add, subtract, multiply). $\sin x$, $e^x$, $\ln x$ are not. Approximation lets us replace difficult functions with quick polynomial computations.

Taylor's idea: force the polynomial and $f$ to agree on all derivatives at a chosen point $a$. The two then track each other very closely near $a$.

🎵 Audio 1 — Why polynomial approximation? — audio/ch6_01_motivation.mp3

6.2 Deriving the Taylor Coefficients

Suppose
$$f(x) = c_0 + c_1(x-a) + c_2(x-a)^2 + c_3(x-a)^3 + \cdots$$

Set $x = a$:
$$f(a) = c_0 \Rightarrow c_0 = f(a)$$

Differentiate both sides:
$$f'(x) = c_1 + 2c_2(x-a) + 3c_3(x-a)^2 + \cdots$$

Set $x = a$: $f'(a) = c_1 \Rightarrow c_1 = f'(a)$.

Differentiate again:
$$f''(x) = 2c_2 + 6 c_3(x-a) + \cdots$$

Set $x = a$: $f''(a) = 2 c_2 \Rightarrow c_2 = \frac{f''(a)}{2}$.

After $n$ differentiations and setting $x = a$:

$$f^{(n)}(a) = n! \cdot c_n \Rightarrow \boxed{c_n = \frac{f^{(n)}(a)}{n!}}$$

🎵 Audio 2 — Coefficient derivation — audio/ch6_02_derivation.mp3

6.3 Definitions

Taylor series of $f$ centered at $a$:

$$T(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$$

Taylor polynomial of degree $n$ (partial sum):

$$P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x - a)^k$$

Maclaurin series: a Taylor series centered at $a = 0$:

$$M(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$$

🔑 Caution: a Taylor series can converge but not to $f(x)$. Equality requires the Lagrange remainder to vanish (Chapter 8).

🎵 Audio 3 — Taylor / Maclaurin definitions — audio/ch6_03_definitions.mp3

6.4 Maclaurin Series from the Definition

Example 1: $f(x) = e^x$

All derivatives equal $e^x$, so $f^{(n)}(0) = 1$.

$$M(x) = \sum_{n=0}^{\infty} \frac{1}{n!} x^n = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

Radius of convergence $R = \infty$ (verified in Ch. 7).

Example 2: $f(x) = \sin x$

Derivatives cycle with period 4: $\sin, \cos, -\sin, -\cos$.

At $x = 0$: $0, 1, 0, -1, 0, 1, \dots$

Only odd-order coefficients survive, and signs alternate:

$$M(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$

Example 3: $f(x) = \cos x$

Cycle: $\cos, -\sin, -\cos, \sin$. At 0: $1, 0, -1, 0, 1, \dots$.

$$M(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$$

🎵 Audio 4 — Three classics: $e^x$, $\sin$, $\cos$ — audio/ch6_04_three_classics.mp3

6.5 Expanding About $a \ne 0$

Find the Taylor series of $f(x) = \ln x$ at $a = 1$.

$f(1) = 0$, $f'(x) = \frac{1}{x}$ so $f'(1) = 1$, $f''(x) = -\frac{1}{x^2}$ so $f''(1) = -1$, $f'''(x) = \frac{2}{x^3}$ so $f'''(1) = 2$.

In general $f^{(n)}(1) = (-1)^{n-1}(n-1)!$ for $n \ge 1$.

$$T(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(n-1)!}{n!}(x-1)^n = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}(x-1)^n$$

🔑 Strategy: compute the first 3–4 derivatives until you spot the pattern, then write the closed form.

6.6 Numerical Estimation with Taylor Polynomials

Example

Estimate $\sin(0.1)$ using the 4th-degree Maclaurin polynomial.

$$P_4(x) = x - \frac{x^3}{6}$$

(Note: $\sin$ has zero even-order coefficients, so degree 4 is effectively the cubic.)

$P_4(0.1) = 0.1 - \frac{0.001}{6} \approx 0.0998333$.

True value $\sin(0.1) \approx 0.0998334$, error $< 10^{-6}$. ✅

🎵 Audio 5 — Numerical estimation — audio/ch6_05_approximation.mp3

6.7 Trap Alerts ⚠️

1. Don't drop the $n!$ in the coefficient $\frac{f^{(n)}(a)}{n!}$.
2. "Degree-$n$ Taylor polynomial" means the highest power is $n$ — but functions like $\sin$ may have zero coefficients on even powers.
3. Convergence ≠ equality with $f$. You need the Lagrange remainder to vanish.
4. The center $a$ is not always 0. AP loves $a = 1$ or $a = \pi/4$.
5. First few derivatives are easy to miscompute — slow down on the first 3–4, then generalize.

6.8 Mnemonic

"Function = polynomial sum, coefficients are $f^{(n)}(a)/n!$"

• Maclaurin = Taylor at $a = 0$
• Memorize three classics: $e^x$, $\sin$, $\cos$
• Compute first 3–4 derivatives, then generalize

🎵 Audio 6 — Recap + mnemonic — audio/ch6_06_recap.mp3

Media Inventory

End of chapter. Next: common Maclaurin expansions and operations.