Key Taylor Series Expansions

Why This Matters

Taylor series are the bridge between the calculus you've learned—derivatives, integrals, limits—and the powerful idea that any smooth function can be rebuilt from its derivatives at a single point. On the BC exam, you're being tested on whether you understand how polynomials can approximate transcendental functions, why those approximations work, and how to quantify their accuracy. This isn't just abstract theory: Taylor series let you evaluate impossible integrals, solve differential equations, and understand function behavior in ways that direct computation simply can't.

The key concepts you'll encounter include polynomial approximation, convergence and divergence, error bounds, and series manipulation. Don't just memorize the expansions for $e^x$, $\sin x$, and $\cos x$—know why they take the forms they do (alternating signs, factorial denominators, odd vs. even powers). When you understand the underlying structure, you can derive any series on the spot and tackle FRQ questions that ask you to build, manipulate, or analyze Taylor polynomials.

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The Foundation: Taylor and Maclaurin Polynomials

Taylor polynomials capture a function's behavior by matching its value and derivatives at a center point. The more derivatives you match, the better your approximation becomes near that point.

Taylor Series Definition

• General form: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$—each term encodes one derivative's contribution
• Coefficient formula $\frac{f^{(n)}(a)}{n!}$ ensures the polynomial's $n$th derivative at $x = a$ matches $f^{(n)}(a)$
• Polynomial approximation means replacing complicated functions with sums of powers—the foundation of numerical analysis

Maclaurin Series

• Special case where $a = 0$, giving the simpler form $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$
• Computational advantage—evaluating derivatives at zero often yields clean integer or zero values
• Most common expansions ($e^x$, $\sin x$, $\cos x$, $\ln(1+x)$) are Maclaurin series because their centers at zero produce elegant patterns

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The Essential Expansions You Must Know

These four series appear constantly on the BC exam. Memorize their forms, but understand their patterns—odd/even powers, alternating signs, and factorial growth.

Exponential Function $e^x$

• Expansion: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}$
• All positive terms—since every derivative of $e^x$ equals $e^x$, and $e^0 = 1$
• Converges for all real $x$ (infinite radius of convergence), making it the most well-behaved series

Sine Function $\sin x$

• Expansion: $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$
• Odd powers only—reflects that $\sin x$ is an odd function ($\sin(-x) = -\sin x$)
• Alternating signs come from the derivative cycle: $\sin \to \cos \to -\sin \to -\cos$

Cosine Function $\cos x$

• Expansion: $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$
• Even powers only—reflects that $\cos x$ is an even function ($\cos(-x) = \cos x$)
• Same derivative cycle as sine produces the alternating pattern; converges for all real $x$

Natural Logarithm $\ln(1+x)$

• Expansion: $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n}$
• No factorials—coefficients are simply $\frac{1}{n}$, making this series grow more slowly
• Limited convergence: only valid for $-1 < x \leq 1$—a common exam trap

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Error Analysis and the Lagrange Remainder

Understanding error bounds separates students who can use Taylor series from those who truly understand them. The Lagrange remainder tells you exactly how wrong your approximation might be.

Taylor's Theorem

• Core guarantee—any sufficiently smooth function equals its Taylor polynomial plus a remainder term
• Truncation creates error: stopping at degree $n$ means ignoring infinitely many higher-order terms
• Foundation for approximation—the theorem justifies why Taylor polynomials work at all

Lagrange Remainder Form

• Formula: $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ for some $c$ between $a$ and $x$
• The mystery $c$—you don't know its exact value, so you bound $|f^{(n+1)}(c)|$ by its maximum on the interval
• Error bound strategy: find $M$ such that $|f^{(n+1)}(c)| \leq M$, then $|R_n(x)| \leq \frac{M \cdot |x-a|^{n+1}}{(n+1)!}$

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Convergence: Where Does the Series Work?

A Taylor series isn't useful if it doesn't converge. The radius of convergence defines the "safe zone" where your series actually represents the function.

Radius of Convergence

• Definition: the value $R$ such that the series converges for $|x - a| < R$ and diverges for $|x - a| > R$
• Ratio test is your primary tool: if $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = L$, then $R = \frac{1}{L}$
• Three possibilities: $R = 0$ (converges only at center), $R = \infty$ (converges everywhere), or finite $R$

Interval of Convergence

• More than just radius—you must check endpoints separately using other convergence tests
• Endpoint behavior varies: $\ln(1+x)$ converges at $x = 1$ but diverges at $x = -1$
• Common exam question: find the interval, then determine whether each endpoint is included

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Building and Manipulating Series

You won't always be handed a series—sometimes you need to construct one or modify a known series. These techniques turn memorized formulas into flexible problem-solving tools.

Finding Coefficients

• Coefficient formula: $a_n = \frac{f^{(n)}(a)}{n!}$ requires computing derivatives at the center
• Pattern recognition—after finding the first few coefficients, look for a general formula
• Derivative tables on FRQs often provide $f(a), f'(a), f''(a), \ldots$ so you can build the polynomial directly

Series Operations

• Addition/subtraction: combine term by term when series have the same center
• Multiplication: use the Cauchy product or multiply out first few terms for polynomial approximations
• Substitution: replace $x$ with another expression (e.g., substitute $-x^2$ into $e^x$ to get $e^{-x^2}$)

Composition and Substitution

• Most efficient technique—rather than computing derivatives of $e^{-x^2}$, substitute $-x^2$ into the known series for $e^u$
• Convergence changes: substituting $x^2$ for $x$ in $\ln(1+x)$ gives $\ln(1+x^2)$, valid for $|x| \leq 1$
• Integration/differentiation of series: integrate or differentiate term by term within the interval of convergence

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Power Series Connection

Taylor series are a specific type of power series. Understanding this relationship helps you see Taylor series as part of a larger framework.

Relationship to Power Series

• Every Taylor series is a power series of the form $\sum_{n=0}^{\infty} c_n(x-a)^n$
• Not every power series is Taylor—Taylor series have coefficients determined by derivatives; general power series don't require this
• Uniqueness theorem: if a function has a power series representation centered at $a$, it must be the Taylor series

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Quick Reference Table

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Self-Check Questions

1. Which two series among $e^x$, $\sin x$, $\cos x$, and $\ln(1+x)$ share the property of converging for all real numbers? What makes them different from $\ln(1+x)$?

2. If you're given a table of $f(2)$, $f'(2)$, $f''(2)$, and $f'''(2)$, write the formula for the third-degree Taylor polynomial centered at $x = 2$.

3. Compare and contrast the series for $\sin x$ and $\cos x$: How do their terms reflect the odd/even nature of each function?

4. You need to approximate $e^{0.1}$ with error less than $0.0001$. How would you use the Lagrange remainder to determine how many terms are sufficient?

5. Explain why substituting $-x^2$ into the series for $e^x$ is more efficient than computing derivatives of $e^{-x^2}$ directly. What is the resulting series for $e^{-x^2}$?