Key Concepts of Power Series

Why This Matters

Power series are one of the most powerful tools in Calculus II because they let you transform complicated functions into infinite polynomials that you can differentiate, integrate, and manipulate with ease. You're being tested on your ability to determine where a series converges, how to represent functions as series, and why term-by-term operations work. These concepts connect directly to approximation theory, differential equations, and the fundamental relationship between discrete sums and continuous functions.

Don't just memorize formulas—understand the underlying logic. When you see a power series problem, you should immediately think about convergence behavior, the relationship between a function and its series representation, and how operations like differentiation and integration affect the series. Master these conceptual categories, and you'll handle any power series question the exam throws at you.

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Foundational Structure

Power series have a specific anatomy that determines everything else—their convergence, their usefulness, and how you manipulate them. Understanding the basic form is essential before tackling applications.

Definition of a Power Series

• General form $\sum_{n=0}^{\infty} a_n (x - c)^n$—where $a_n$ are coefficients and $c$ is the center point
• Function representation—expresses a function as an infinite sum of polynomial terms built on powers of $(x - c)$
• Conditional convergence—the series converges for some $x$ values and diverges for others, which is why determining the domain is critical

Geometric Series

• Simplest power series form $\sum_{n=0}^{\infty} ar^n$—where $a$ is the first term and $r$ is the common ratio
• Convergence condition—converges only when $|r| < 1$, diverges when $|r| \geq 1$
• Closed-form sum $\frac{a}{1 - r}$—this formula appears constantly in series manipulation problems and provides a template for more complex series

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Convergence Analysis

The central question for any power series is where does it converge? This determines the domain where your series actually represents a meaningful function.

Radius of Convergence

• Definition—the value $R$ that defines how far from center $c$ the series converges, found using $R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}}$
• Convergence rule—series converges when $|x - c| < R$ and diverges when $|x - c| > R$
• Special cases—$R = 0$ means convergence only at the center; $R = \infty$ means convergence everywhere

Interval of Convergence

• Beyond the radius—the interval includes all $x$ values where the series converges, typically $(c - R, c + R)$
• Endpoint testing required—the radius tells you the interior, but you must test $x = c - R$ and $x = c + R$ separately using other convergence tests
• Notation matters—use parentheses for excluded endpoints, brackets for included ones (e.g., $[c - R, c + R)$)

Convergence Tests for Power Series

• Ratio Test—evaluate $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$; if less than 1, the series converges absolutely
• Root Test—evaluate $\lim_{n \to \infty} \sqrt[n]{|a_n|}$; particularly useful when coefficients involve $n$th powers
• Endpoint tests—once you find $R$, apply alternating series test, p-series test, or comparison tests at the boundary points

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Function Representation

Power series aren't just abstract objects—they represent real functions. Taylor and Maclaurin series formalize how to build a power series from any sufficiently smooth function.

Taylor Series

• Definition—represents a function as $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x - c)^n$ using derivatives at center $c$
• Coefficient formula—each coefficient $a_n = \frac{f^{(n)}(c)}{n!}$ encodes the function's behavior through its derivatives
• Approximation power—truncating the series gives polynomial approximations; more terms means better accuracy near $c$

Maclaurin Series

• Special case—a Taylor series centered at $c = 0$, giving $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$
• Common series to memorize—$e^x$, $\sin x$, $\cos x$, $\frac{1}{1-x}$, and $\ln(1+x)$ all have standard Maclaurin forms
• Computational efficiency—functions evaluated at zero often simplify dramatically, making Maclaurin series the go-to choice when possible

Representation of Functions as Power Series

• Wide applicability—exponential, trigonometric, logarithmic, and rational functions can all be expressed as power series
• Problem-solving strategy—convert difficult functions to series form, then use polynomial techniques for integration or differentiation
• Differential equations—power series solutions work when standard methods fail, making this representation essential for advanced applications

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Operations on Power Series

One of the most powerful features of power series is that you can differentiate and integrate them term by term—as long as you stay within the interval of convergence.

Differentiation of Power Series

• Term-by-term rule—the derivative of $\sum_{n=0}^{\infty} a_n (x - c)^n$ is $\sum_{n=1}^{\infty} n a_n (x - c)^{n-1}$
• Radius preserved—differentiation doesn't change the radius of convergence (though endpoint behavior may change)
• Application—use this to find series representations of derivatives or to verify that a series satisfies a differential equation

Integration of Power Series

• Term-by-term rule—the integral of $\sum_{n=0}^{\infty} a_n (x - c)^n$ is $\sum_{n=0}^{\infty} \frac{a_n}{n+1} (x - c)^{n+1} + C$
• Radius preserved—like differentiation, integration maintains the same radius of convergence
• Application—this lets you find antiderivatives of functions that have no elementary closed form, like $e^{-x^2}$

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Quick Reference Table

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Self-Check Questions

1. What is the relationship between the radius of convergence and the interval of convergence, and why must you test endpoints separately?

2. Compare and contrast Taylor series and Maclaurin series—when would you choose one over the other for approximating a function?

3. If you differentiate a power series, what happens to its radius of convergence? What about its interval of convergence?

4. Which convergence test would you use for a series with coefficients involving factorials versus one with coefficients involving $n$th powers?

5. Given the Maclaurin series for $\frac{1}{1-x}$, how would you derive the series for $\frac{1}{(1-x)^2}$ and for $\ln(1-x)$ using term-by-term operations?