Study Guide: Taylor and Maclaurin Series

Learning Objectives

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

• Describe the procedure for finding a Taylor polynomial of a given order for a function.
• Explain the meaning and significance of Taylor's theorem with remainder.
• Estimate the remainder (the error) for a Taylor series approximation of a given function.

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.
• Taylor Series: A power series expansion of a function $f(x) centered around a specific point x=a$.
• Maclaurin Series: A specific type of Taylor series where the center of expansion is exactly $x=0$.
• Taylor Polynomial: A truncated (finite) version of a Taylor series used to approximate a function.
• Taylor's Theorem with Remainder: A mathematical rule that quantifies the exact difference (error) between a function and its Taylor polynomial approximation.
• Interval of Convergence: The specific range of $x$-values for which a power series successfully converges to a finite value.

The "Big Idea"

Many essential mathematical functions, such as $e^x$, $\sin(x)$, and $\ln(x), cannot be calculated using simple arithmetic. The **"Big Idea"** behind Taylor and Maclaurin series is that we can represent these complex, non-polynomial functions as **infinite polynomials** (power series). Because polynomials are simple to differentiate, integrate, and compute, transforming a complex function into a Taylor series unlocks our ability to solve differential equations, evaluate non-elementary integrals, and allow calculators to compute values like \sin(0.12)$ rapidly and accurately.

[!NOTE] A Taylor series perfectly mimics a function by matching its value, its slope (first derivative), its concavity (second derivative), and all higher-order derivatives at a single specific anchor point $x=a$.

Formula / Concept Box

Hierarchical Outline

• 1. Power Series Foundations
  • Definition: Treat functions as infinite sequences of polynomial terms.
  • Calculus of Series: Power series can be differentiated and integrated term-by-term.
• 2. Constructing the Series
  • Taylor Series: Anchoring the polynomial at $x=a by matching all derivatives f^{(n)}(a)$.
  • Maclaurin Series: The simplified case anchoring at $x=0$, widely used for core functions.
• 3. Approximation and Error Analysis
  • Taylor Polynomials: Cutting off the infinite series at degree $n$ to create a practical approximation.
  • Taylor's Theorem: Establishing that $f(x) = P_n(x) + R_n(x)$.
  • Remainder Estimation: Bounding the maximum possible value of the next derivative to guarantee precision.

Visual Anchors

Process of Finding a Taylor Series

Polynomial Approximation of $e^x$

[!TIP] Notice in the graph above how higher-degree polynomials ($P_2$ vs $P_1) "hug" the actual function curve (e^x) for a wider interval around the center a=0$.

Definition-Example Pairs

• Term: Taylor Polynomial
  • Definition: A finite sum of the first $n$ terms of a Taylor series.
  • Real-World Example: Like a digital camera's resolution; a low-degree polynomial is a pixelated, rough outline of the function, while a high-degree polynomial provides a high-definition, highly accurate match.
• Term: Taylor's Remainder ($R_n$)
  • Definition: The mathematical expression for the exact difference between the true function and the $n$-th degree Taylor polynomial.
  • Real-World Example: Like a store receipt that tells you exactly how much "change" (error) you have left over after paying with an approximation.
• Term: Interval of Convergence
  • Definition: The specific set of $x$-values where the infinite polynomial successfully adds up to the original function.
  • Real-World Example: Like a Bluetooth connection range; the polynomial perfectly syncs with the true function inside the range, but breaks down into garbage data if you step outside of it.

Worked Examples

▶Example 1: Finding a Maclaurin Polynomial

Problem: Find the 3rd-degree Maclaurin polynomial $P_3(x)$ for $f(x) = \sin(x)$.

Step 1: Find the function and its first 3 derivatives.

• $f(x) = \sin(x)$
• $f'(x) = \cos(x)$
• $f''(x) = -\sin(x)$
• $f'''(x) = -\cos(x)$

Step 2: Evaluate these at the center $a=0$.

• $f(0) = \sin(0) = 0$
• $f'(0) = \cos(0) = 1$
• $f''(0) = -\sin(0) = 0$
• $f'''(0) = -\cos(0) = -1$

Step 3: Plug into the Maclaurin formula. $P_3(x) = 0 + \frac{1}{1!}x + \frac{0}{2!}x^2 + \frac{-1}{3!}x^3$

Final Answer: $P_3(x) = x - \frac{x^3}{6}$

▶Example 2: Using Taylor's Remainder Theorem

Problem: Use the 2nd-degree Maclaurin polynomial for $e^x$ to approximate $e^{0.1}$ and use Taylor's Inequality to estimate the maximum error.

Step 1: Recall the polynomial. For $f(x)=e^x$, $P_2(x) = 1 + x + \frac{x^2}{2}$. Approximation: $P_2(0.1) = 1 + 0.1 + \frac{0.01}{2} = 1.105$.

Step 2: Set up Taylor's Inequality for $n=2$. $|R_2(x)| \leq \frac{M}{(3)!} |x|^3$ Where $M is the maximum value of |f'''(c)|$ on the interval $[0, 0.1]$.

Step 3: Find M. The third derivative is $e^x$. On $[0, 0.1]$, $e^x is increasing, so its max is at e^{0.1}. Since we don't know e^{0.1}, we can safely say e^{0.1} < e^1 < 3$. Let $M=3$.

Step 4: Calculate the bound. $|R_2(0.1)| \leq \frac{3}{6} (0.1)^3 = 0.5 \times 0.001 = 0.0005$

Conclusion: The approximation $1.105 is accurate to within $0.0005 of the true value of $e^{0.1}$.

Checkpoint Questions

1. What is the main difference between a Taylor series and a Maclaurin series?
2. Why do we need the remainder term $R_n(x)$ when using Taylor polynomials?
3. If you increase the degree $n of a Taylor polynomial, what generally happens to the error of the approximation near the center a$?
4. What function is represented by the Maclaurin series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$?

[!WARNING] Common Pitfall: Don't forget the factorial $n! in the denominator of the Taylor series formula! A very common mistake is simply writing \sum f^{(n)}(a)(x-a)^n$, which will lead to vastly incorrect approximations.