Taylor Series Approximation

A Taylor series expands a smooth function $f$ around a point $a$ as an infinite polynomial: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$$ When $a = 0$, this is called a Maclaurin series. Intuition: You're building a polynomial that matches $f$ at $a$ in value, slope, curvature, and every higher derivative. Each term corrects the next order of approximation. For points close to $a$, even a few terms give excellent accuracy. Essential expansions you should know: - $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ - $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ for $|x| < 1$ - $\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots$ for $|x| < 1$ - $(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \cdots$ (binomial series) Concrete example: Approximate $e^{0.1}$. Using two terms: $e^{0.1} \approx 1 + 0.1 + 0.$$\frac{01}{2}$ = 1.105. Exact value: $1.10517...$ — error of 0.02%. When to use: Approximating complicated expressions (especially in finance: $\ln(1+r) \approx r$ for small $r$). Deriving results in probability (moment generating functions). Simplifying integrals. The backbone of perturbation methods. This is a fundamental technique — arguably the single most useful technique in applied mathematics.