Taylor & Maclaurin Series

Part 1 of 7 — Taylor Polynomial Definition

Taylor Series (centered at $x = a$)

f(x) = sum_{n=0}^{infty} rac{f^{(n)}(a)}{n!}(x-a)^n

Maclaurin Series (centered at $a = 0$)

f(x) = sum_{n=0}^{infty} rac{f^{(n)}(0)}{n!}x^n = f(0) + f'(0)x + rac{f''(0)}{2!}x^2 + cdots

$n$th-Degree Taylor Polynomial

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

Taylor Basics 🎯

Key Takeaways — Part 1

Taylor: expand around $a$. Maclaurin: expand around $0$. Coefficient of $(x-a)^n$ is $f^{(n)}(a)/n!$.

Taylor & Maclaurin Series

Part 2 of 7 — The Big Four Maclaurin Series

Must-Know Series

e^x = sum_{n=0}^{infty} rac{x^n}{n!} = 1 + x + rac{x^2}{2!} + rac{x^3}{3!} + cdots quad (|x| < infty)

sin x = sum_{n=0}^{infty} rac{(-1)^n x^{2n+1}}{(2n+1)!} = x - rac{x^3}{3!} + rac{x^5}{5!} - cdots quad (|x| < infty)

cos x = sum_{n=0}^{infty} rac{(-1)^n x^{2n}}{(2n)!} = 1 - rac{x^2}{2!} + rac{x^4}{4!} - cdots quad (|x| < infty)

rac{1}{1-x} = sum_{n=0}^{infty} x^n = 1 + x + x^2 + x^3 + cdots quad (|x| < 1)

Big Four 🎯

Key Takeaways — Part 2

Memorize $e^x$, $\sin x$, $\cos x$, $\frac{1}{1-x}$. Build all others from these four!

Taylor & Maclaurin Series

Part 3 of 7 — Building New Series

Substitution

e^{x^2} = sum rac{(x^2)^n}{n!} = sum rac{x^{2n}}{n!}

Multiplication

xsin x = xleft(x - rac{x^3}{6} + cdots ight) = x^2 - rac{x^4}{6} + cdots

Differentiation

rac{d}{dx}left(rac{1}{1-x} ight) = rac{1}{(1-x)^2} = sum_{n=1}^{infty} nx^{n-1}

Integration

int rac{1}{1+x^2},dx = arctan x = sum_{n=0}^{infty} rac{(-1)^n x^{2n+1}}{2n+1}

Building Series 🎯

Key Takeaways — Part 3

Substitution, multiplication, differentiation, and integration all apply term-by-term.

Taylor & Maclaurin Series

Part 4 of 7 — Taylor Series at $a eq 0$

Taylor Series Centered at $a$

f(x) = sum_{n=0}^{infty}rac{f^{(n)}(a)}{n!}(x-a)^n

Example: $ln x$ about $a = 1$

$f(x) = ln x$, $f(1) = 0$ $f'(x) = 1/x$, $f'(1) = 1$ $f''(x) = -1/x^2$, $f''(1) = -1$

ln x = (x-1) - rac{(x-1)^2}{2} + rac{(x-1)^3}{3} - cdots = sum_{n=1}^{infty}rac{(-1)^{n+1}(x-1)^n}{n}

Non-Zero Center 🎯

Key Takeaways — Part 4

Non-zero center: compute derivatives at $x = a$, use $(x-a)^n$.

Taylor & Maclaurin Series

Part 5 of 7 — Using Series to Evaluate Limits & Integrals

Limits via Taylor Series

lim_{x o 0}rac{sin x - x}{x^3} = lim_{x o 0}rac{(x - x^3/6 + cdots) - x}{x^3} = lim_{x o 0}rac{-x^3/6 + cdots}{x^3} = -rac{1}{6}

Integrals via Taylor Series

$int_0^1 e^{-x^2},dx$: no elementary antiderivative!

e^{-x^2} = 1 - x^2 + rac{x^4}{2} - rac{x^6}{6} + cdots

int_0^1 = left[x - rac{x^3}{3} + rac{x^5}{10} - rac{x^7}{42} + cdots ight]_0^1 = 1 - 1/3 + 1/10 - 1/42 + cdots

Series Applications 🎯

Key Takeaways — Part 5

Taylor series elegantly evaluate limits and integrals that are otherwise difficult.

Taylor Series

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Taylor & Maclaurin Series — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Taylor & Maclaurin Series — Complete! ✅