Glossary

13.4 Applications of Taylor Series

4 min read•july 30, 2024

are powerful tools for approximating functions using polynomials. They let us represent complex functions as infinite sums of simpler terms, making calculations easier. This technique is crucial for estimating function values, integrals, and derivatives in mathematical analysis.

Applications of Taylor series extend beyond simple approximations. They're used to solve differential equations, estimate errors in calculations, and even model physical phenomena. Understanding these applications helps bridge the gap between theoretical concepts and practical problem-solving in advanced mathematics.

Taylor series approximations

Representing functions as infinite series

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  • Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point
  • The general form of a Taylor series for a function f(x)f(x) about a point aa is:
    • f(x)=f(a)+f′(a)(x−a)+(f′′(a)/2!)(x−a)2+(f′′′(a)/3!)(x−a)3+...f(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + (f'''(a)/3!)(x-a)^3 + ...
  • is a special case of Taylor series where a=0a = 0
  • Common Maclaurin series include:
    • ex=1+x+(x2/2!)+(x3/3!)+...e^x = 1 + x + (x^2/2!) + (x^3/3!) + ...
    • sin(x)=x−(x3/3!)+(x5/5!)−...sin(x) = x - (x^3/3!) + (x^5/5!) - ...
    • cos(x)=1−(x2/2!)+(x4/4!)−...cos(x) = 1 - (x^2/2!) + (x^4/4!) - ...

Taylor polynomials and approximations

  • Taylor polynomials are finite approximations of Taylor series, obtained by truncating the series after a certain number of terms
  • The degree of a is the highest power of (x−a)(x-a) included in the approximation
  • Taylor series can be used to:
    • Approximate functions near the point of expansion
    • Solve problems involving complicated functions
    • Find limits of functions
  • The accuracy of the approximation depends on the number of terms used and the proximity to the point of expansion

Taylor series for estimation

Estimating function values

  • To estimate the value of a function at a point using Taylor series:
    • Substitute the point into the Taylor series expansion
    • Evaluate the resulting expression
  • The accuracy of the estimation depends on:
    • The number of terms used in the Taylor polynomial
    • The proximity of the point to the center of expansion
  • Example: Estimating sin(0.1)sin(0.1) using the Maclaurin series for sin(x)sin(x)
    • sin(0.1)≈0.1−(0.13/3!)+(0.15/5!)≈0.0998sin(0.1) ≈ 0.1 - (0.1^3/3!) + (0.1^5/5!) ≈ 0.0998

Estimating integrals and derivatives

  • Taylor series can be integrated or differentiated term by term to estimate values of integrals or derivatives
  • When integrating or differentiating Taylor series, the may change and needs to be considered
  • Taylor series approximations can be used to estimate definite integrals by integrating the Taylor polynomial over the given interval
  • Example: Estimating ∫01exdx∫_0^1 e^x dx using the Maclaurin series for exe^x
    • ∫01exdx≈∫01(1+x+(x2/2!)+(x3/3!))dx≈1.7183∫_0^1 e^x dx ≈ ∫_0^1 (1 + x + (x^2/2!) + (x^3/3!)) dx ≈ 1.7183

Limitations of Taylor series

Interval of convergence

  • Taylor series approximations are valid only within the interval of convergence
    • The interval of convergence is the range of xx-values for which the series converges to the original function
  • The interval of convergence can be determined using:
    • The ratio test
    • Other methods from the study of series convergence
  • Example: The Maclaurin series for 1/(1−x)1/(1-x) has an interval of convergence of (−1,1)(-1, 1)

Error analysis and Lagrange error bound

  • The error in a Taylor polynomial approximation is the difference between the actual function value and the approximation
    • The error is represented by the remainder term
  • The Lagrange error bound gives an upper bound for the absolute value of the remainder term
    • It is based on the maximum value of the next higher-order derivative on the interval between the point of expansion and the point of estimation
  • As the degree of the Taylor polynomial increases:
    • The approximation becomes more accurate
    • The computational complexity also increases
  • Example: The Lagrange error bound for the nn-th degree Taylor polynomial of exe^x about x=0x=0 is ∣Rn(x)∣≤∣x∣(n+1)/(n+1)!|R_n(x)| ≤ |x|^(n+1)/(n+1)!

Functions not representable by Taylor series

  • Some functions cannot be represented by a Taylor series due to the lack of derivatives at the point of expansion
  • Example: e(−1/x2)e^(-1/x^2) at x=0x=0 does not have a Taylor series representation because it is not differentiable at x=0x=0

Power series solutions for differential equations

Assuming a power series solution

  • Power series methods involve assuming a solution to a differential equation in the form of a power series
    • The coefficients of the series are determined by substituting the series into the differential equation and equating coefficients
  • The general form of a power series solution is:
    • y=∑(n=0to∞)cn(x−a)ny = ∑(n=0 to ∞) c_n (x-a)^n, where cnc_n are the coefficients to be determined and aa is the point around which the series is centered

Solving differential equations using power series

  • To solve a differential equation using power series:
    1. Assume a solution in the form of a power series with unknown coefficients
    2. Substitute the power series and its derivatives into the differential equation
    3. Simplify and equate coefficients of like powers of (x−a)(x-a) to obtain a recurrence relation for the coefficients
    4. Use the recurrence relation and initial conditions to determine the coefficients of the power series solution
  • The interval of convergence for the power series solution can be found using the ratio test or other methods
    • The solution is valid only within this interval

Applications of power series methods

  • Power series methods can be used to solve:
    • Linear differential equations with variable coefficients
    • Some nonlinear differential equations
  • Example: Solving the differential equation y′−xy=0y' - xy = 0 with y(0)=1y(0) = 1 using power series
    • Assume y=∑(n=0to∞)cnxny = ∑(n=0 to ∞) c_n x^n, substitute into the equation, and equate coefficients to find the recurrence relation c(n+1)=cn/(n+1)c_(n+1) = c_n/(n+1)
    • Using y(0)=1y(0) = 1, the solution is y=1+x+(x2/2!)+(x3/3!)+...y = 1 + x + (x^2/2!) + (x^3/3!) + ..., which converges for all xx

Key Terms to Review (17)

Approximation of functions: Approximation of functions is a mathematical technique used to find simpler expressions that closely resemble more complex functions over a certain interval. This approach is essential in various analyses as it allows for easier computation and understanding of function behavior. Key methods for approximation include polynomial approximations, notably through Taylor series, which provide a way to express functions as an infinite sum of terms calculated from the values of their derivatives at a single point.
Augustin-Louis Cauchy: Augustin-Louis Cauchy was a French mathematician whose work laid the groundwork for modern analysis, particularly in the study of limits, continuity, and integrals. His contributions, including the formalization of the concept of a limit and the development of the Riemann integral, have had a profound impact on mathematical analysis and are foundational to various important results and theorems.
Binomial Series: The binomial series is an infinite series that provides a way to express powers of binomials in terms of simpler terms, particularly when dealing with expressions like $(a + b)^n$. It can be represented using the binomial coefficient and is an extension of the binomial theorem, allowing for expansion even when $n$ is not a non-negative integer. This series is particularly useful in approximating functions and analyzing polynomial behavior.
Brook Taylor: Brook Taylor was an English mathematician known for his contributions to calculus, particularly Taylor's Theorem and Taylor Series. His work provided a powerful method for approximating functions using polynomials, connecting ideas of derivatives and limits in a way that deepened the understanding of mathematical analysis and laid the groundwork for further developments in series expansions.
Differentiation: Differentiation is a fundamental concept in calculus that refers to the process of finding the derivative of a function, which measures how the function's output changes as its input changes. It helps to understand the rate of change, slopes of curves, and is essential for analyzing functions, optimizing problems, and modeling real-world scenarios. The principles of differentiation not only apply in pure mathematical contexts but also have significant implications in various fields such as physics, engineering, and economics.
Error Analysis: Error analysis is the study of the types and sources of errors in mathematical approximations and computations. It focuses on quantifying how inaccuracies affect results, especially when using methods like Taylor series to estimate functions. By understanding error analysis, one can assess the reliability of approximations and improve mathematical modeling.
Exponential function: An exponential function is a mathematical function of the form $$f(x) = a imes b^{x}$$, where $a$ is a constant, $b$ is a positive real number, and $x$ is the variable. Exponential functions model growth or decay processes and are characterized by their constant percentage rate of change. These functions are vital for understanding limits, series expansions, and their applications in real-world scenarios.
Integration: Integration is a fundamental concept in mathematics that refers to the process of finding the integral of a function, which can be understood as the reverse operation of differentiation. It allows for the calculation of areas under curves, accumulation of quantities, and solutions to various problems across different fields. This concept is pivotal in analyzing continuous functions and is closely tied to the idea of limits and summation, establishing a foundation for further applications and implications in mathematical analysis.
Interval of convergence: The interval of convergence is the set of all values for which a power series converges. This concept is critical when working with power series and Taylor series, as it helps determine the range of inputs for which the series produces valid results. Understanding this interval allows for the application of various tests for convergence and provides insight into the behavior of functions represented by these series.
Lagrange Remainder Theorem: The Lagrange Remainder Theorem provides a way to express the error or remainder of approximating a function by its Taylor series. It states that the remainder of the Taylor series for a function at a point can be represented as a specific term involving the (n+1)th derivative of the function, evaluated at some point between the center of expansion and the point of interest. This theorem is crucial in understanding how accurately a Taylor series can approximate a function, which is particularly useful in applications involving Taylor series.
Maclaurin Series: A Maclaurin series is a special case of the Taylor series centered at zero, representing a function as an infinite sum of terms calculated from the values of its derivatives at that point. This series is useful for approximating functions using polynomials, which can simplify calculations and provide insights into function behavior near the origin. The series can be applied in various mathematical contexts, revealing important properties of functions and facilitating numerical analysis.
Radius of convergence: The radius of convergence is a measure that determines the interval within which a power series converges to a finite value. It is essential for understanding the behavior of power series and provides insight into their convergence properties, which are crucial when applying them to functions, approximations, and series tests.
Taylor Polynomial: A Taylor polynomial is an approximation of a function using the derivatives of that function at a single point. It provides a way to express a function as an infinite sum of terms calculated from the values of its derivatives at that point, which can be truncated to create a polynomial that closely resembles the function near that point. This concept is vital in approximating complex functions and analyzing their behavior in various applications.
Taylor Series: A Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point. This powerful tool allows us to approximate functions with polynomials, facilitating easier analysis and computation across various contexts. The connection between Taylor series and power series broadens their utility, enabling convergence analysis and revealing the behavior of functions in specified intervals.
Taylor's Theorem: Taylor's Theorem provides a way to approximate a function using polynomials, specifically by expressing a function as an infinite sum of its derivatives evaluated at a specific point. This theorem is foundational for understanding how functions behave locally and serves as the basis for deriving Taylor and Maclaurin series, which are used to represent functions in calculus. By utilizing the Mean Value Theorem, Taylor's Theorem demonstrates the relationship between derivatives and the behavior of functions near a point, leading to applications in various fields through the use of Taylor Series.
Trigonometric Functions: Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides, primarily used in geometry and analysis. They include sine, cosine, tangent, and their reciprocals, cosecant, secant, and cotangent, and they are fundamental in various applications such as modeling periodic phenomena and analyzing waveforms. These functions are also continuous and periodic, making them essential in calculus and series expansions.
σ: In mathematical analysis, the symbol σ often represents a measure or a summation index in various contexts, indicating a systematic approach to aggregating values. This term is crucial in understanding Riemann sums, where σ can denote partition indices, and it also plays a significant role in series convergence, such as when discussing Taylor series and their coefficients. Additionally, σ is linked to the conditions under which certain integrals converge uniformly, highlighting its versatility across different areas of mathematical analysis.
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