Glossary

12.4 Functions as Power Series

3 min read•august 7, 2024

Functions as power series are a powerful tool in calculus. They let us represent complex functions as infinite sums of simpler terms. This technique bridges the gap between polynomials and more advanced functions, opening up new ways to analyze and work with them.

In this part of the chapter, we'll see how to express common functions like exponentials and as power series. We'll also learn about Taylor and , which approximate functions near specific points using their derivatives.

Taylor and Maclaurin Series

Approximating Functions with Polynomials

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  • Taylor polynomials approximate a function f(x)f(x) near a point aa by using the function's derivatives at that point
    • The nth-degree Taylor polynomial is denoted as Pn(x)P_n(x)
    • Formula: Pn(x)=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+...+f(n)(a)n!(x−a)nP_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n
  • Maclaurin series is a special case of where a=0a=0
    • Formula: Pn(x)=f(0)+f′(0)x+f′′(0)2!x2+...+f(n)(0)n!xnP_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + ... + \frac{f^{(n)}(0)}{n!}x^n
  • Examples:
    • Taylor polynomial for f(x)=exf(x)=e^x at a=0a=0 (Maclaurin): Pn(x)=1+x+x22!+x33!+...+xnn!P_n(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... + \frac{x^n}{n!}
    • Taylor polynomial for f(x)=sin⁡xf(x)=\sin x at a=π2a=\frac{\pi}{2}: Pn(x)=1−(x−π2)22!+(x−π2)44!−...P_n(x) = 1 - \frac{(x-\frac{\pi}{2})^2}{2!} + \frac{(x-\frac{\pi}{2})^4}{4!} - ...

Error Bounds and Remainder Term

  • states that any smooth function can be represented by a Taylor series plus a remainder term Rn(x)R_n(x)
    • Formula: f(x)=Pn(x)+Rn(x)f(x) = P_n(x) + R_n(x)
  • The remainder term represents the error between the function and its Taylor polynomial approximation
    • As nn increases, the remainder term decreases, improving the approximation
  • Error bounds provide an upper limit for the absolute value of the remainder term
    • Helps determine how many terms are needed for a desired level of accuracy
    • Example: For f(x)=exf(x)=e^x, the error bound is ∣Rn(x)∣≤M(n+1)!∣x−a∣n+1|R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1}, where MM is the maximum value of ∣f(n+1)(t)∣|f^{(n+1)}(t)| on the interval between aa and xx

Important Series Expansions

Exponential Function Series

  • The exe^x can be represented by its Maclaurin series
    • Formula: ex=1+x+x22!+x33!+...+xnn!+...e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... + \frac{x^n}{n!} + ...
    • Converges for all real values of xx
  • Other exponential functions can be derived from this series
    • Example: e2x=1+2x+(2x)22!+(2x)33!+...+(2x)nn!+...e^{2x} = 1 + 2x + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + ... + \frac{(2x)^n}{n!} + ...

Trigonometric Function Series

  • The sine and cosine functions have Maclaurin series representations
    • Sine series: sin⁡x=x−x33!+x55!−x77!+...\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...
    • Cosine series: cos⁡x=1−x22!+x44!−x66!+...\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...
    • Both series converge for all real values of xx
  • These series can be used to derive other trigonometric function series
    • Example: tan⁡x=sin⁡xcos⁡x=x+x33+2x515+...\tan x = \frac{\sin x}{\cos x} = x + \frac{x^3}{3} + \frac{2x^5}{15} + ...

Binomial Series

  • The is a generalization of the binomial theorem for non-integer exponents
    • Formula: (1+x)r=1+rx+r(r−1)2!x2+r(r−1)(r−2)3!x3+...(1+x)^r = 1 + rx + \frac{r(r-1)}{2!}x^2 + \frac{r(r-1)(r-2)}{3!}x^3 + ...
    • Converges for ∣x∣<1|x| < 1 and all real values of rr
  • Special cases include the (r=−1)(r=-1) and the square root series (r=12)(r=\frac{1}{2})
    • Example: (1+x)12=1+12x−18x2+116x3−...(1+x)^{\frac{1}{2}} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - ...

Key Terms to Review (18)

Analytic functions: Analytic functions are functions that are locally represented by convergent power series. This means that around any point in their domain, these functions can be expressed as an infinite sum of terms calculated from the function's derivatives at that point. The ability to represent a function as a power series not only highlights its smoothness but also reveals essential properties such as differentiability and the existence of Taylor series expansions.
Approximation of Functions: Approximation of functions refers to the process of estimating the values of a function using simpler or more easily computable functions, often through techniques like power series and Taylor series. This approach allows for easier calculations and better understanding of the function's behavior near a specific point or over an interval. It plays a crucial role in both theoretical analysis and practical applications, particularly in numerical methods and solving differential equations.
Binomial Series: The binomial series is an infinite series expansion that represents the function $(1 + x)^n$ for any real number $n$. This series allows us to express functions as a sum of terms involving powers of $x$, which can be particularly useful for approximating functions and understanding their behavior near certain points. It highlights the connection between algebraic expressions and power series, showing how we can expand them into an infinite series.
Cauchy-Hadamard Theorem: The Cauchy-Hadamard Theorem provides a way to determine the radius of convergence for power series. It states that the radius of convergence, denoted as R, can be found using the formula $$\frac{1}{R} = \limsup_{n \to \infty} \sqrt[n]{|a_n|}$$ where \(a_n\) are the coefficients of the power series. This theorem is crucial for understanding where a given power series converges or diverges, thus playing a fundamental role in the study of functions represented as power series.
Error estimation: Error estimation is the process of determining the possible error or uncertainty in a computed value or approximation. This concept is crucial when using series expansions, as it helps quantify how closely the approximation aligns with the actual function. By understanding error estimation, one can assess the reliability of approximations made through power series, allowing for better decision-making in mathematical analysis.
Exponential Function: An exponential function is a mathematical expression in the form $$f(x) = a \cdot b^{x}$$, where 'a' is a constant, 'b' is a positive real number, and 'x' is the variable exponent. This type of function exhibits rapid growth or decay and is fundamental in modeling real-world phenomena such as population growth, radioactive decay, and compound interest. Exponential functions are closely related to logarithmic functions, allowing for conversions between exponential and logarithmic forms.
Function representation: Function representation is the way in which a mathematical function is expressed or depicted, often using various forms such as equations, graphs, or power series. This concept is crucial as it allows for a deeper understanding of how functions behave and how they can be analyzed and manipulated in different contexts.
Geometric Series: A geometric series is the sum of the terms of a geometric sequence, which is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Geometric series can converge to a finite limit or diverge to infinity depending on the value of the common ratio, making them a crucial concept in understanding infinite series and their convergence behavior. This concept also extends to power series, where functions can be expressed as sums of geometric series, providing important insights into their behavior around specific points.
Interval of convergence: The interval of convergence refers to the set of values for which a power series converges to a finite limit. It is crucial in understanding where a series can be used to represent functions accurately, as it dictates the domain over which the power series is valid. The endpoints of this interval may or may not be included, depending on whether the series converges at those points.
Maclaurin Series: A Maclaurin series is a special case of the Taylor series that represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point, specifically at zero. This series allows us to express functions in terms of their derivatives, providing a powerful tool for approximating functions near the origin and analyzing their behavior.
N-th term: The n-th term refers to the general term of a sequence or series, expressed in terms of its position 'n'. This concept is crucial for understanding infinite series, as it allows for the identification of specific terms and aids in convergence tests. By analyzing the n-th term, one can determine the behavior of the entire sequence or series as 'n' approaches infinity.
Radius of convergence: The radius of convergence is the distance from the center of a power series within which the series converges to a finite value. This concept is crucial when determining the interval over which a power series represents a function, as it indicates where the series reliably approximates the function's value. Understanding this radius helps identify where power series can be effectively used for calculations and approximations.
Taylor Series: A Taylor series is an infinite sum of terms that represents a function as a power series, where the coefficients are derived from the function's derivatives at a specific point. This concept is crucial for approximating functions with polynomials and helps in understanding the behavior of functions near that point, connecting various mathematical ideas like convergence, power series, and applications in calculus.
Taylor's Theorem: Taylor's Theorem is a fundamental principle in calculus that provides an approximation of a function as a sum of its derivatives at a specific point. This theorem connects the concept of derivatives with power series, allowing for the expression of functions as infinite series, facilitating easier computation and analysis.
Term-by-term differentiation: Term-by-term differentiation is a technique used to differentiate power series by applying the differentiation operator to each term individually. This process allows for the derivation of new power series from an original series, preserving the structure and properties of the series within a specified interval of convergence.
Term-by-term integration: Term-by-term integration is a method used to integrate power series by integrating each term of the series individually. This technique is essential for working with power series, as it allows us to find the integral of a series within its interval of convergence, preserving the properties of the series as it converges to a function.
Trigonometric Functions: Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides, commonly used to model periodic phenomena. These functions include sine, cosine, tangent, and their inverses, and they play a crucial role in various fields, including physics, engineering, and computer science.
σ (sigma notation): Sigma notation is a concise way to represent the sum of a sequence of numbers, using the Greek letter sigma (σ) to indicate summation. It allows mathematicians and students to express complex summations in a compact form, facilitating easier manipulation and evaluation of series, especially when working with functions represented as power series.
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