Study Guide: Working with Taylor Series

Learning Objectives

By the end of this study guide, you should be able to:

• Write the terms of the binomial series for various powers.
• Recognize the Taylor and Maclaurin series expansions of common, elementary functions.
• Apply algebraic and calculus techniques (substitution, differentiation, integration) to find the Taylor series for a new function.
• Use Taylor series to solve differential equations.
• Evaluate nonelementary integrals by representing the integrand as a power series.

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Key Terms & Glossary

• Power Series: An infinite series of the form $\sum c_n(x-a)^n that acts as an "infinite polynomial," representing a function within its interval of convergence.
• Taylor Series: A specific power series representation of a function f(x) centered at a point a, constructed using the function's derivatives at a.
• Maclaurin Series: A special case of a Taylor series where the center of the series is at a = 0.
• Binomial Series: The Maclaurin series expansion of the function (1+x)^k, which generalizes the binomial theorem to fractional and negative powers.
• Nonelementary Integral: An integral (like \int e^{-x^2} dx) whose antiderivative cannot be written in terms of basic elementary functions (polynomials, exponentials, logarithms, trigonometric functions).

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The "Big Idea"

[!IMPORTANT] The core purpose of Working with Taylor Series is to translate complex, hard-to-manipulate functions into "infinite polynomials." Because polynomials are the easiest mathematical objects to differentiate and integrate, expressing functions as Taylor or Maclaurin series unlocks our ability to evaluate nonelementary integrals, solve stubborn differential equations, and approximate complex real-world behaviors with simple arithmetic.

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Formula / Concept Box

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Hierarchical Outline

• 1. Finding Taylor Series
  • Direct Computation: Using the definition and calculating $f^{(n)}(a) iteratively.
  • Substitution: Plugging a modified argument (e.g., x^2) into a known Maclaurin series.
  • Multiplication/Division: Multiplying a known series by a polynomial (e.g., x^2 e^x).
• 2. The Binomial Series
  • Fractional Exponents: Expanding square roots like \sqrt{1+x}.
  • Negative Exponents: Expanding rational functions like \frac{1}{(1+x)^2}.
• 3. Applying Taylor Series in Calculus
  • Nonelementary Integrals: Replacing the integrand with its series and integrating term-by-term.
  • Differential Equations: Assuming a series solution y = \sum c_n x^n$ and solving for coefficients.

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Visual Anchors

1. Strategy for Finding a Taylor Series

2. Convergence of Taylor Polynomials

The more terms we use, the closer the Taylor polynomial models the original function.

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Definition-Example Pairs

• Term-by-Term Integration

  • Definition: The process of taking the integral of a power series one term at a time, just like a standard polynomial.
  • Real-World Example: In statistics, the "bell curve" probability involves evaluating $\int e^{-x^2} dx. Since this is nonelementary, statisticians use term-by-term integration of its Taylor series to calculate standard deviations and probabilities.

• Binomial Series

  • Definition: The infinite series representation of (1+x)^k$where$k is any real number.
  • Real-World Example: Physicists use the binomial series expansion of \gamma = (1 - v^2/c^2)^{-1/2} to approximate relativistic kinetic energy at everyday speeds, yielding the familiar \frac{1}{2}mv^2.

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Worked Examples

▶Example 1: Evaluating a Nonelementary Integral

Problem: Evaluate \int \sin(x^2) dx as an infinite series.

Step 1: Start with the known series. We know the Maclaurin series for \sin(u)$ is: $\sin(u) = u - \frac{u^3}{3!} + \frac{u^5}{5!} - \frac{u^7}{7!} + \dots$

Step 2: Substitute $u = x^2$. $\sin(x^2) = (x^2) - \frac{(x^2)^3}{3!} + \frac{(x^2)^5}{5!} - \dots$ $\sin(x^2) = x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!} - \dots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{4n+2}}{(2n+1)!}$

Step 3: Integrate term-by-term. $\int \sin(x^2) dx = \int \left( x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!} - \dots \right) dx$ $= C + \frac{x^3}{3} - \frac{x^7}{7 \cdot 3!} + \frac{x^{11}}{11 \cdot 5!} - \dots$ $= C + \sum_{n=0}^{\infty} (-1)^n \frac{x^{4n+3}}{(4n+3)(2n+1)!}$

▶Example 2: Using the Binomial Series

Problem: Find the first three non-zero terms of the Maclaurin series for $f(x) = \frac{1}{\sqrt{1-x}}$.

Step 1: Rewrite the function to match the binomial form. $f(x) = (1 + (-x))^{-1/2}$ Here, our $k = -1/2 and our variable is (-x)$.

Step 2: Apply the binomial series formula. $(1+u)^k \approx 1 + ku + \frac{k(k-1)}{2!}u^2$

Step 3: Substitute $u = -x$ and compute coefficients. $1 + \left(-\frac{1}{2}\right)(-x) + \frac{(-\frac{1}{2})(-\frac{3}{2})}{2}(-x)^2$ $= 1 + \frac{1}{2}x + \frac{3}{8}x^2$

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Checkpoint Questions

1. What is the difference between a Taylor series and a Maclaurin series?
2. Why is it usually faster to use substitution on a known Maclaurin series than to compute the derivatives directly from the definition?
3. State the first four non-zero terms of the Maclaurin series for $e^x.
4. How does expressing an integrand as a Taylor series allow you to evaluate a nonelementary integral?
5. When applying the binomial series to (1+x)^k, under what condition does the series converge?

[!TIP] Self-Check Answers: (1) Maclaurin is simply a Taylor series centered at a=0. (2) Direct computation requires calculating n$derivatives, which can quickly become complex via the product/chain rules. Substitution relies on basic algebra. (3) $1 + x + x^2/2 + x^3/6$. (4) You can integrate the series term-by-term using the basic power rule for integration. (5) It converges for $|x| < 1$.