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IB Mathematics AA – Topic 5: Calculus

Comprehensive Guide to Maclaurin Series

Introduction to Maclaurin Series

Maclaurin series are powerful tools that represent functions as infinite polynomials centered at zero. They transform complex transcendental functions like exponentials, logarithms, and trigonometric functions into elegant polynomial approximations that can be manipulated using basic algebra. From calculating approximations to solving differential equations, Maclaurin series provide both theoretical insight and practical computational methods.

Key concepts: A Maclaurin series expands a function as an infinite sum of terms involving increasing powers of x, where each coefficient is determined by the function's derivatives at x = 0. When truncated (shortened), these series provide polynomial approximations that become more accurate with more terms. Standard functions have well-known expansions that can be memorized and manipulated through substitution, multiplication, differentiation, and integration.

Why Maclaurin series matter: These series enable approximation of complicated functions using simple polynomials, making calculations feasible in engineering, physics, and computer science. They simplify integration of functions that lack elementary antiderivatives, solve differential equations that resist standard techniques, and provide deep insight into function behavior near zero. Understanding series convergence and manipulation is essential for advanced mathematics and scientific computing.

In this guide: We'll master the standard Maclaurin series expansions for key functions, apply substitution techniques to find series for composite functions, multiply series to find products, differentiate and integrate series term-by-term, solve differential equations using series methods, and tackle comprehensive IB HL exam problems—all essential skills for exam success.

1. Standard Maclaurin Series Expansions

The General Formula

Maclaurin Series General Form

\(f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots\)

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

Given in IB formula booklet

Essential Expansions to Memorize

Standard Functions (Given in Formula Booklet)

Exponential Function

\(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\)

Valid for all x

Sine Function

\(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\)

Odd powers only; valid for all x

Cosine Function

\(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\)

Even powers only; valid for all x

Natural Logarithm

\(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots\)

Valid for \(-1 < x \leq 1\)

Binomial Expansion

\((1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots\)

Valid for \(|x| < 1\) when n not a positive integer

⚠ Common Pitfalls:

  • Factorials matter: Don't forget denominators (2!, 3!, etc.)
  • Sign patterns: Watch alternating signs in sin, cos, ln series
  • Convergence: ln(1+x) only valid for \(-1 < x \leq 1\)
  • Powers: Sin has odd powers; cos has even powers

2. Using Substitution

Substitution Method

Strategy for Composite Functions:

  1. Identify the "inside function" (what to substitute)
  2. Start with standard series for "outside function"
  3. Replace every x with the inside function
  4. Expand and simplify to required number of terms

Example: For \(e^{2x}\), substitute \(2x\) into series for \(e^x\)

💡 Substitution Tips:

  • Always specify "up to and including \(x^n\) term"
  • Include extra terms before substituting—you can discard later
  • Simplify coefficients completely
  • Watch for powers increasing rapidly with substitution

Example 1: Substitution Method

Problem: Find the Maclaurin series for \(\sin(3x)\) up to and including the \(x^5\) term

Solution:

Step 1: Start with standard series

\(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\)

Step 2: Substitute \(3x\) for \(x\)

\(\sin(3x) = (3x) - \frac{(3x)^3}{3!} + \frac{(3x)^5}{5!} - \cdots\)

Step 3: Expand and simplify

\(= 3x - \frac{27x^3}{6} + \frac{243x^5}{120}\)

\(= 3x - \frac{9x^3}{2} + \frac{81x^5}{40}\)

\(\sin(3x) = 3x - \frac{9x^3}{2} + \frac{81x^5}{40}\)

Note: Each power of 3 multiplies the coefficient

3. Products of Maclaurin Series

Multiplying Series

Method for Products:

  1. Write out each series to sufficient number of terms
  2. Put each series in brackets
  3. Multiply brackets, keeping only terms up to required power
  4. Collect like terms (same powers of x)
  5. Simplify coefficients

Key: Include enough terms so product gives all needed powers

Example 2: Product of Series (IB-Style)

Problem: Find the Maclaurin series for \(e^x \cos x\) up to and including the \(x^3\) term

Solution:

Step 1: Write individual series

\(e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots\)

\(\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots\)

For \(x^3\) term, need up to \(x^3\) in first series and \(x^2\) in second

Step 2: Multiply

\(e^x \cos x = \left(1 + x + \frac{x^2}{2} + \frac{x^3}{6}\right)\left(1 - \frac{x^2}{2}\right)\)

Step 3: Expand (keep terms up to \(x^3\))

\(= 1 - \frac{x^2}{2} + x - \frac{x^3}{2} + \frac{x^2}{2} - \frac{x^4}{4} + \frac{x^3}{6} - \frac{x^5}{12}\)

Discard \(x^4\) and \(x^5\) terms

Step 4: Collect like terms

Constant: \(1\)

\(x\) term: \(x\)

\(x^2\) term: \(-\frac{x^2}{2} + \frac{x^2}{2} = 0\)

\(x^3\) term: \(-\frac{x^3}{2} + \frac{x^3}{6} = \frac{-3x^3 + x^3}{6} = -\frac{2x^3}{6} = -\frac{x^3}{3}\)

\(e^x \cos x = 1 + x - \frac{x^3}{3}\)

4. Differentiation and Integration of Series

Term-by-Term Differentiation

Differentiate Each Term:

If \(f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots\)

Then \(f'(x) = a_1 + 2a_2x + 3a_3x^2 + 4a_4x^3 + \cdots\)

Rule: Differentiate term-by-term using power rule

Constant term vanishes; each power decreases by 1

Term-by-Term Integration

Integrate Each Term:

If \(f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots\)

Then \(\int f(x)dx = C + a_0x + \frac{a_1x^2}{2} + \frac{a_2x^3}{3} + \frac{a_3x^4}{4} + \cdots\)

Rule: Integrate term-by-term using power rule

Don't forget constant of integration \(C\)

Applications

Common Uses:

  • Verifying series: Differentiate \(e^x\) series to get itself
  • Finding new series: Integrate \(\frac{1}{1+x}\) to get \(\ln(1+x)\)
  • Approximating integrals: Integrate series for functions without elementary antiderivatives
  • Solving DEs: Use series methods when standard techniques fail

5. Maclaurin Series and Differential Equations

Series Solutions to DEs

Using Series to Solve DEs:

  1. Assume solution has form \(y = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots\)
  2. Differentiate term-by-term to find \(y'\), \(y''\), etc.
  3. Substitute into differential equation
  4. Match coefficients of like powers of x
  5. Solve for unknown coefficients using initial conditions

Verification of Solutions

Common Verification Problems:

  • Show that \(e^x\) series satisfies \(\frac{dy}{dx} = y\)
  • Verify \(\sin x\) and \(\cos x\) series satisfy \(\frac{d^2y}{dx^2} = -y\)
  • Demonstrate relationships between function and its derivatives

📋 Maclaurin Series Quick Reference

Function Maclaurin Series Validity
\(e^x\) \(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\) All x
\(\sin x\) \(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\) All x
\(\cos x\) \(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\) All x
\(\ln(1+x)\) \(x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\) \(-1 < x \leq 1\)
\((1+x)^n\) \(1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots\) \(|x| < 1\)

🎯 IB Exam Strategy

Common Question Types:

  • "Find Maclaurin series up to \(x^n\)": Use formula booklet expansions
  • "Using substitution...": Replace x in standard series, expand and simplify
  • "Find series for product...": Multiply series in brackets, collect terms
  • "Differentiate the series...": Differentiate term-by-term
  • "Hence find approximation...": Substitute small value of x
  • "Verify solution to DE...": Differentiate series and substitute into equation

Key Reminders:

  • Standard series are in formula booklet—use them!
  • Always state "up to and including \(x^n\)" in answer
  • Include extra terms before substitution or multiplication
  • Simplify all coefficients completely
  • Check convergence conditions for ln and binomial series
  • For products: multiply brackets, keep relevant powers only

🎉 Master Maclaurin Series!

Maclaurin series represent one of mathematics' most elegant ideas—expressing complex functions as infinite polynomials. From approximating calculations to solving differential equations that resist standard methods, series provide both theoretical insight and practical computational power. Master the standard expansions, substitution techniques, and manipulation methods to excel in IB HL exams and prepare for advanced mathematics!

Key Success Factors:

  • ✓ Know standard series: \(e^x\), \(\sin x\), \(\cos x\), \(\ln(1+x)\), \((1+x)^n\)
  • ✓ Substitution: replace every x in standard series
  • ✓ Products: multiply in brackets, collect like terms
  • ✓ Differentiate/integrate term-by-term
  • ✓ Include enough terms BEFORE simplifying
  • ✓ Always state convergence conditions for ln and binomial

Use Formula Booklet • Substitute Carefully • Simplify Completely

Master Maclaurin series and excel in IB Mathematics! 🚀