Series approximations (HL: Taylor/Maclaurin) - Mathematics: Analysis & Approaches IB Study Notes

Taylor and Maclaurin series provide polynomial approximations of complex functions, fundamental to IB Mathematics AA HL Topic 5.

Maclaurin Series (special case where a = 0): $$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ... + \frac{f^{(n)}(0)}{n!}x^n + ...$$

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

Taylor Series (centred at x = a): $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$

Key Standard Series (memorize these):

• Exponential: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$
• Sine: $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...$ (odd powers, alternating signs)
• Cosine: $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...$ (even powers, alternating signs)
• Natural logarithm: $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...$ for $|x| < 1$

Mnemonic: "Every Student Can Learn" = Exponential, Sine, Cosine, Logarithm (the four essential series)

Radius of convergence determines where the series accurately represents the function. The remainder term $R_n(x)$ measures approximation error.

Cambridge Definition: A Taylor series is an infinite sum representing a function as the sum of terms calculated from the function's derivatives at a single point.

Think of Taylor series as mathematical zoom lenses—the more terms you include, the clearer your "picture" of the function becomes near a specific point.

The Building Block Analogy: Imagine constructing a curved slide using straight planks. One plank (linear approximation) is crude. Two planks (quadratic) follow the curve better. More planks create increasingly accurate approximations. Each derivative adds another "plank" that captures more curvature detail.

Real-World Applications:

1. Physics & Engineering: Calculating pendulum motion uses $\sin \theta \approx \theta - \frac{\theta^3}{6}$ for small angles, simplifying differential equations dramatically.

2. Computer Graphics: Video games and animation software use Taylor series to approximate trigonometric functions quickly—calculating $\sin x$ exactly takes too long, but a 5-term approximation runs instantly.

3. Financial Mathematics: Options pricing models (Black-Scholes) use series expansions to approximate complex exponential and logarithmic functions for rapid calculations.

4. Space Navigation: NASA uses Taylor series to approximate spacecraft trajectories, balancing computational speed with precision.

Why Centre at Different Points? If you want accurate approximations near $x = 5$, centre your Taylor series at $a = 5$ rather than using Maclaurin (centred at 0). It's like setting up camp near your hiking destination rather than miles away.

Convergence Insight: Some series converge everywhere ($e^x$), while others have limited ranges ($\ln(1+x)$ only for $|x| < 1$). This is like radio signals—some broadcast globally, others only locally.