13.1 Definition and Properties of Power Series

Power series are like building blocks for functions, letting us represent complex mathematical relationships as infinite sums. They're super useful for approximating functions and solving tricky equations. Understanding power series is key to grasping advanced calculus concepts.

In this section, we'll dive into the nuts and bolts of power series. We'll learn how to define them, identify their parts, and perform basic operations. Plus, we'll explore how to determine when and where these series converge.

Power Series Structure

Definition and Representation

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• Power series are infinite series of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$, where:
  • $a_n$ are the coefficients that determine the magnitude of each term
  • $c$ is the center, which represents the point around which the series is expanded
  • $x$ is the variable
• Power series represent functions as an infinite sum of terms involving powers of a variable
  • Examples of functions that can be represented by power series include polynomials ($x^2 + 2x + 1$), exponential functions ($e^x$), and trigonometric functions (sin($x$), cos($x$))

General Term and Domain of Convergence

• The general term of a power series is $a_n(x-c)^n$, where $n$ is a non-negative integer
  • This term determines the structure of each individual component in the infinite series
• The domain of convergence of a power series is the set of all values of $x$ for which the series converges
  • Outside the domain of convergence, the series may diverge or have undefined behavior
  • The domain of convergence can be determined using various convergence tests

Center and Coefficients of Power Series

Identifying the Center

• The center of a power series is the value $c$ in the general term $a_n(x-c)^n$
  • It represents the point around which the series is expanded
• To identify the center, look for the value of $x$ that is being subtracted from the variable in each term
  • For example, in the series $\sum_{n=0}^{\infty} (x-3)^n$, the center is $c=3$

Identifying the Coefficients

• The coefficients of a power series are the values $a_n$ in the general term $a_n(x-c)^n$
  • They determine the magnitude of each term in the series
• To identify the coefficients, look for the constant values multiplied by each power of $(x-c)$
  • For example, in the series $\sum_{n=0}^{\infty} 2^n(x-1)^n$, the coefficients are $a_n=2^n$
• The center and coefficients can be used to determine properties such as the radius of convergence and behavior near the center

Arithmetic Operations with Power Series

Addition of Power Series

• Power series with the same center can be added term by term, provided that the resulting series converges
• To add two power series, add the coefficients of like powers of $(x-c)$
  • For example, $\sum_{n=0}^{\infty} a_n(x-c)^n + \sum_{n=0}^{\infty} b_n(x-c)^n = \sum_{n=0}^{\infty} (a_n+b_n)(x-c)^n$
• The resulting power series will have the same center as the original series

Multiplication of Power Series

• Power series can be multiplied using the Cauchy product
  • Multiply each term of one series by each term of the other and collect like powers of $(x-c)$
• The Cauchy product of two power series $\sum_{n=0}^{\infty} a_n(x-c)^n$ and $\sum_{n=0}^{\infty} b_n(x-c)^n$ is given by $\sum_{n=0}^{\infty} c_n(x-c)^n$, where $c_n = \sum_{k=0}^{n} a_kb_{n-k}$
• The resulting power series will have the same center as the original series
• The radius of convergence of the product will be at least the minimum of the radii of convergence of the original series

Convergence of Power Series

Convergence at a Specific Point

• A power series converges at a specific point $x=x_0$ if the sequence of partial sums $S_n(x_0) = \sum_{k=0}^{n} a_k(x_0-c)^k$ approaches a finite limit as $n$ approaches infinity
• Convergence at a specific point can be determined using various tests:
  • Ratio test: If $\lim_{n \to \infty} |\frac{a_{n+1}(x_0-c)^{n+1}}{a_n(x_0-c)^n}| < 1$, the series converges at $x=x_0$. If the limit is greater than 1, the series diverges at $x=x_0$
  • Root test: If $\lim_{n \to \infty} \sqrt[n]{|a_n(x_0-c)^n|} < 1$, the series converges at $x=x_0$. If the limit is greater than 1, the series diverges at $x=x_0$
  • Comparison test: Compare the terms of a power series with the terms of a known convergent or divergent series to determine convergence

Convergence Behavior

• If a power series converges at a specific point, it will converge for all points closer to the center
• If a power series diverges at a specific point, it will diverge for all points farther from the center
• The set of all points where a power series converges is called the interval of convergence
  • The interval of convergence is always centered around the center of the power series
  • The radius of convergence is the distance from the center to the endpoints of the interval of convergence