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Calculus - Maclaurin and Taylor Series

Grade 12IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Definition of Taylor Series: A Taylor series is a way to represent a smooth function f(x)f(x) as an infinite sum of polynomial terms calculated from the values of its derivatives at a specific point x=ax = a. Visually, the first term f(a)f(a) represents a horizontal line, the second term adds the slope (tangent line), and higher-order terms adjust the curvature (concavity) to 'hug' the original function's graph more closely around the center aa.

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The Maclaurin Series: This is a special case of the Taylor series where the expansion is centered at x=0x = 0. In IB Mathematics, many problems focus on Maclaurin series because they simplify calculations. Graphically, a Maclaurin series approximation is most accurate at the yy-axis and loses accuracy as you move further away in either the positive or negative xx direction.

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Convergence and the Radius of Convergence: Not every power series converges to the function for all values of xx. The 'Radius of Convergence' RR defines the distance from the center aa within which the series accurately represents the function. On a graph, within the interval (a−R,a+R)(a-R, a+R), the polynomial and the function are virtually indistinguishable as nn increases, but outside this interval, the polynomial series will diverge sharply away from the function.

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Symmetry and Parity: The Maclaurin series of even functions, such as cos⁡(x)\cos(x), contain only even powers of xx, reflecting their symmetry across the yy-axis. Similarly, odd functions like sin⁡(x)\sin(x) contain only odd powers, reflecting their rotational symmetry about the origin. Recognizing these patterns helps in identifying errors and simplifying series for trigonometric functions.

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Series Manipulation via Substitution: Instead of calculating derivatives from scratch, you can create new series by substituting expressions into known standard series. For example, to find the series for ex2e^{x^2}, you simply replace every xx in the exe^x series with x2x^2. Visually, this transforms the standard exponential curve into a symmetric, bell-shaped approximation centered at x=0x=0.

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Term-by-Term Differentiation and Integration: Power series can be differentiated or integrated term-by-term within their radius of convergence. Differentiating a series term-by-term results in a new series representing the derivative of the original function. Graphically, if the original series approximates a curve, the differentiated series approximates the slope of that curve at every point within the interval of convergence.

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The Role of Factorials: The n!n! in the denominator of Taylor terms ensures that the coefficients of higher-order terms usually decrease rapidly. This mathematical 'damping' is what allows the infinite sum to converge to a finite value. It also means that for small values of (x−a)(x-a), the first few terms provide a very high-quality 'local' approximation of the function.

📐Formulae

General Taylor Series: f(x)=∑n=0∞f(n)(a)n!(x−a)n=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+…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 + \dots

General Maclaurin Series: f(x)=∑n=0∞f(n)(0)n!xn=f(0)+f′(0)x+f′′(0)2!x2+…f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots

Exponential Series: ex=1+x+x22!+x33!+⋯=∑n=0∞xnn!e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots = \sum_{n=0}^{\infty} \frac{x^n}{n!}

Sine Series: sin⁡(x)=x−x33!+x55!−⋯=∑n=0∞(−1)nx2n+1(2n+1)!\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}

Cosine Series: cos⁡(x)=1−x22!+x44!−⋯=∑n=0∞(−1)nx2n(2n)!\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}

Natural Logarithm Series: ln⁡(1+x)=x−x22+x33−⋯=∑n=1∞(−1)n+1xnn\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} for −1<x≤1-1 < x \leq 1

Geometric Series: 11−x=1+x+x2+x3+⋯=∑n=0∞xn\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots = \sum_{n=0}^{\infty} x^n for ∣x∣<1|x| < 1

Binomial Series: (1+x)k=1+kx+k(k−1)2!x2+…(1+x)^k = 1 + kx + \frac{k(k-1)}{2!}x^2 + \dots for ∣x∣<1|x| < 1

💡Examples

Problem 1:

Find the Maclaurin series for f(x)=cos⁡(2x)f(x) = \cos(2x) up to and including the term in x4x^4.

Solution:

Step 1: Start with the standard Maclaurin series for cos⁡(u)\cos(u): cos⁡(u)=1−u22!+u44!−…\cos(u) = 1 - \frac{u^2}{2!} + \frac{u^4}{4!} - \dots Step 2: Substitute u=2xu = 2x into the series: cos⁡(2x)=1−(2x)22!+(2x)44!\cos(2x) = 1 - \frac{(2x)^2}{2!} + \frac{(2x)^4}{4!} Step 3: Simplify the powers and factorials: cos⁡(2x)=1−4x22+16x424\cos(2x) = 1 - \frac{4x^2}{2} + \frac{16x^4}{24} Step 4: Final simplified form: cos⁡(2x)=1−2x2+23x4\cos(2x) = 1 - 2x^2 + \frac{2}{3}x^4

Explanation:

This approach uses the substitution method, which is much faster than taking multiple derivatives. By replacing the variable in a known series, we ensure the relationship between terms remains consistent with the original function's symmetry.

Problem 2:

Determine the Taylor series for f(x)=1xf(x) = \frac{1}{x} centered at a=1a = 1, up to the third-order term.

Solution:

Step 1: Calculate the derivatives at x=1x = 1. f(1)=11=1f(1) = \frac{1}{1} = 1 f′(x)=−x−2  ⟹  f′(1)=−1f'(x) = -x^{-2} \implies f'(1) = -1 f′′(x)=2x−3  ⟹  f′′(1)=2f''(x) = 2x^{-3} \implies f''(1) = 2 f′′′(x)=−6x−4  ⟹  f′′′(1)=−6f'''(x) = -6x^{-4} \implies f'''(1) = -6 Step 2: Plug these into the Taylor formula ∑f(n)(1)n!(x−1)n\sum \frac{f^{(n)}(1)}{n!}(x-1)^n: f(x)≈10!(x−1)0+−11!(x−1)1+22!(x−1)2+−63!(x−1)3f(x) \approx \frac{1}{0!}(x-1)^0 + \frac{-1}{1!}(x-1)^1 + \frac{2}{2!}(x-1)^2 + \frac{-6}{3!}(x-1)^3 Step 3: Simplify the coefficients: f(x)≈1−(x−1)+(x−1)2−(x−1)3f(x) \approx 1 - (x-1) + (x-1)^2 - (x-1)^3

Explanation:

For centers other than zero (a≠0a \neq 0), we must use the general Taylor formula. This example demonstrates how the derivatives determine the coefficients, and the (x−a)(x-a) terms shift the 'focus' of the approximation to the point x=1x=1.