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Calculus II
Table of Contents
Unit 6 Overview: Power Series
6.1 Power Series and Functions
6.2 Properties of Power Series
6.3 Taylor and Maclaurin Series
6.4 Working with Taylor Series

➗calculus ii review

6.2 Properties of Power Series

Citation:

Power series are versatile tools in calculus, allowing us to perform various operations like addition, multiplication, and differentiation. These operations help us manipulate and analyze functions represented as infinite sums, expanding our understanding of mathematical relationships.

By mastering power series properties, we gain insights into function behavior and convergence. This knowledge is crucial for solving complex problems in calculus and related fields, enabling us to work with infinite series representations of functions effectively.

Properties of Power Series

Operations on power series

  • Addition and subtraction performed by adding or subtracting corresponding coefficients of like terms
    • Resulting series has the same interval of convergence as the original series (e.g., $\sum_{n=0}^{\infty} a_nx^n + \sum_{n=0}^{\infty} b_nx^n = \sum_{n=0}^{\infty} (a_n + b_n)x^n$)
  • Scalar multiplication achieved by multiplying each coefficient by the scalar value
    • Resulting series has the same interval of convergence as the original series (e.g., $c\sum_{n=0}^{\infty} a_nx^n = \sum_{n=0}^{\infty} (ca_n)x^n$)
  • Uniform convergence of power series within their interval of convergence allows for term-by-term operations

Variable substitution in power series

  • Replace the variable in the original series with a new expression
    • Determine the new interval of convergence based on the substitution (e.g., substituting $x$ with $x-1$ in $\sum_{n=0}^{\infty} a_nx^n$ results in $\sum_{n=0}^{\infty} a_n(x-1)^n$)
  • Multiply each term in the series by a power of the variable
    • Adjust the indices of the summation accordingly (e.g., multiplying $\sum_{n=0}^{\infty} a_nx^n$ by $x^2$ results in $\sum_{n=0}^{\infty} a_nx^{n+2}$)
    • Resulting series may have a different interval of convergence

Multiplication of power series

  • Cauchy product multiplies corresponding terms and sums the results
    • Coefficient of the $n$-th term in the product given by $\sum_{k=0}^{n} a_k b_{n-k}$
    • Resulting series has an interval of convergence at least as large as the intersection of the intervals of convergence of the original series (e.g., $(\sum_{n=0}^{\infty} a_nx^n)(\sum_{n=0}^{\infty} b_nx^n) = \sum_{n=0}^{\infty} (\sum_{k=0}^{n} a_k b_{n-k})x^n$)

Differentiation and integration of series

  • Term-by-term differentiation performed by differentiating each term in the series individually
    • Resulting series has the same interval of convergence as the original series
    • $n$-th term of the differentiated series given by $na_nx^{n-1}$ (e.g., $\frac{d}{dx}\sum_{n=0}^{\infty} a_nx^n = \sum_{n=1}^{\infty} na_nx^{n-1}$)
  • Term-by-term integration achieved by integrating each term in the series individually
    • Resulting series has an interval of convergence at least as large as the original series
    • Constant of integration typically chosen to be 0
    • $n$-th term of the integrated series given by $\frac{a_n}{n+1}x^{n+1}$ (e.g., $\int \sum_{n=0}^{\infty} a_nx^n dx = \sum_{n=0}^{\infty} \frac{a_n}{n+1}x^{n+1} + C$)

Advanced concepts in power series

  • Analytic functions can be represented by power series within their domain of convergence
  • Complex analysis extends the study of power series to the complex plane
  • Laurent series generalize power series to include negative powers of the variable
  • Abel's theorem relates the behavior of a power series at the boundary of its convergence interval to its sum function

Key Terms to Review (31)

Absolute convergence: Absolute convergence occurs when the series $\sum |a_n|$ converges. It implies that the series $\sum a_n$ also converges, regardless of the sign of its terms.
Annuities: Annuities are financial products that provide regular payments over time, typically used for retirement income. Their value can be analyzed using calculus concepts such as series and integrals.
Interval of convergence: The interval of convergence is the set of all real numbers for which a given power series converges. It includes the radius of convergence and specifies whether the endpoints are included or excluded.
Maclaurin series: A Maclaurin series is a special case of the Taylor series, centered at zero. It represents a function as an infinite sum of its derivatives at zero.
Present value: Present value is the current worth of a future sum of money or stream of cash flows given a specified rate of return. It discounts future amounts to reflect their value in today's terms.
Radius of convergence: The radius of convergence is the distance within which a power series converges to a finite value. It determines the interval around the center point where the series is valid.
Ratio test: The Ratio Test is used to determine the convergence or divergence of an infinite series by examining the limit of the ratio of successive terms. It is particularly useful for series with factorials or exponential functions.
Power series: A power series is an infinite series of the form $\sum_{n=0}^{\infty} a_n (x - c)^n$, where $a_n$ represents the coefficient of the nth term and $c$ is a constant. Power series can be used to represent functions within their interval of convergence.
Root test: The root test is a method used to determine the convergence or divergence of an infinite series by examining the nth root of the absolute value of its terms. It provides a useful criterion especially when dealing with series where ratio tests are inconclusive.
Term-by-term differentiation of a power series: Term-by-term differentiation of a power series involves differentiating each term of the series individually. This process is valid within the radius of convergence of the original power series.
Taylor series: A Taylor series is an infinite sum of terms that represents a function as a series of its derivatives evaluated at a single point. The series converges to the function within a certain interval around that point.
Term-by-term integration of a power series: Term-by-term integration of a power series involves integrating each term of the series individually within its interval of convergence. This process results in a new power series that is the integral of the original.
Ratio Test: The ratio test is a method used to determine the convergence or divergence of a series by examining the behavior of the ratio of consecutive terms. It is a powerful tool for analyzing the convergence of infinite series, particularly power series and sequences.
Root Test: The root test is a method used to determine the convergence or divergence of a series by analyzing the behavior of its terms. It is a powerful tool for studying the properties of power series and evaluating the convergence of sequences.
Radius of Convergence: The radius of convergence is a crucial concept in the study of infinite series and power series. It defines the range of values for the independent variable within which the series converges, or in other words, the region where the series can be used to accurately approximate the function it represents.
Power Series: A power series is an infinite series where each term is a variable raised to a non-negative integer power, multiplied by a constant coefficient. Power series are a fundamental concept in calculus, used to represent and analyze functions in a variety of contexts.
Maclaurin Series: A Maclaurin series is a type of Taylor series, which is a power series expansion of a function about its value at a specific point. The Maclaurin series is a special case of the Taylor series where the expansion point is the origin, or $x = 0$. This series is used to approximate and analyze the behavior of functions near the origin.
Term-by-Term Differentiation: Term-by-term differentiation is a technique used to find the derivative of a power series or Taylor series by differentiating each term individually and then combining the results. This method allows for the efficient computation of the derivative of a series without having to differentiate the entire expression as a whole.
Interval of Convergence: The interval of convergence is the range of values of the independent variable for which a power series converges, or in other words, the set of values where the series represents a function. This concept is central to understanding the properties and applications of power series, Taylor series, and Maclaurin series.
Term-by-Term Integration: Term-by-term integration is a method of integrating a power series or Taylor series by integrating each individual term of the series. This technique allows for the integration of complex functions that can be expressed as a series of simpler terms, making the integration process more manageable.
Absolute Convergence: Absolute convergence is a concept in mathematics that describes the behavior of infinite series, where the sum of the absolute values of the series terms converges to a finite value. This property is crucial in understanding the convergence and behavior of various types of series, including alternating series, series involving ratios or roots, and power series.
Abel's Theorem: Abel's Theorem is a fundamental result in the theory of power series that establishes the conditions under which a power series can be integrated or differentiated term-by-term. It provides a rigorous mathematical framework for understanding the properties and behavior of power series, which are essential in the study of calculus and mathematical analysis.
Taylor Series: A Taylor series is a mathematical representation of a function as an infinite sum of terms, each of which is calculated from the values of the function's derivatives at a single point. It is a powerful tool for approximating and analyzing the behavior of functions in the vicinity of a specific point.
Σ (Summation Notation): The summation notation, represented by the Greek letter Σ (sigma), is a mathematical symbol used to represent the sum of a sequence of numbers or terms. It is a concise way to express the addition of multiple values or the evaluation of a function over a range of indices.
Maclaurin: Maclaurin is a type of power series expansion that represents a function as an infinite series around the point $x = 0$. It is a special case of the more general Taylor series expansion, where the function is expanded around an arbitrary point $x = a$.
Taylor: Taylor is a mathematical concept that describes the representation of a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point. It is a powerful tool used in the study of power series and their properties.
Analytic Functions: Analytic functions are a class of functions that can be expressed as a convergent power series in a neighborhood of every point in their domain. They are infinitely differentiable and possess a wide range of mathematical properties that make them valuable tools in various areas of mathematics, including calculus, complex analysis, and differential equations.
Laurent Series: A Laurent series is an infinite series expansion of a complex-valued function that is valid in an annular region around a point, allowing for both positive and negative powers of the variable. It is a generalization of the Taylor series, which is valid only in a disk around the point.
Cauchy Product: The Cauchy product, also known as the convolution product, is a fundamental operation in the theory of power series that allows for the multiplication of two power series. It provides a way to combine the coefficients of two power series to obtain the coefficients of their product.
Complex Analysis: Complex analysis is the study of functions of complex variables, which are mathematical objects that have both real and imaginary components. It builds upon the foundations of real analysis and explores the unique properties and behaviors of complex-valued functions.
Uniform Convergence: Uniform convergence is a concept in mathematical analysis that describes the behavior of a sequence of functions as they converge to a limit function. It is a stronger form of convergence compared to pointwise convergence, ensuring that the convergence is consistent across the entire domain of the functions.