Glossary

13.1 Taylor and Maclaurin Series Expansions

3 min read•august 7, 2024

Taylor and are powerful tools for representing functions as infinite sums. They allow us to approximate complex functions, solve differential equations, and define fundamental mathematical functions like exponentials and .

These series are special types of power series, centered at specific points. The Maclaurin series is a centered at zero, making it a handy special case for many common functions.

Taylor and Maclaurin Series

Defining Taylor and Maclaurin 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
    • Useful for and solving differential equations
    • Can be used to define functions like exponential, trigonometric, and logarithmic functions (exe^x, sin⁡x\sin x, ln⁡(1+x)\ln(1+x))
  • Maclaurin series is a special case of Taylor series centered at x=0x=0
    • are determined by the function's derivatives evaluated at zero
    • Examples include ex=1+x+x22!+x33!+...e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ..., sin⁡x=x−x33!+x55!−...\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...

Power Series and Infinite Series

  • Power series is a series with terms of the form an(x−c)na_n(x-c)^n, where cc is the
    • Coefficients ana_n are constants
    • nn takes on the values 0,1,2,3,...0, 1, 2, 3, ...
  • Infinite series is the sum of an infinite sequence of terms
    • Taylor and Maclaurin series are examples of infinite series
    • Geometric series (1+x+x2+x3+...1 + x + x^2 + x^3 + ...) and harmonic series (1+12+13+14+...1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ...) are other examples

Series Expansion Properties

Center of Expansion and Radius of Convergence

  • Center of expansion is the point x=cx=c around which the Taylor series is developed
    • For Maclaurin series, the center of expansion is always x=0x=0
    • Changing the center of expansion can make the series converge for different values of xx
  • determines the range of xx values for which the series converges
    • Inside the radius of convergence, the series converges to the original function
    • Outside the radius of convergence, the series diverges
    • On the boundary (edge cases), the series may converge or diverge

Taylor Polynomial

  • is a finite sum of terms from a Taylor series, approximating the original function
    • Denoted as Tn(x)T_n(x), where nn is the degree of the polynomial
    • As nn increases, the Taylor polynomial becomes a better approximation of the function
    • Example: The 2nd degree Taylor polynomial for exe^x around x=0x=0 is T2(x)=1+x+x22!T_2(x) = 1 + x + \frac{x^2}{2!}

Error Analysis

Taylor's Theorem and Remainder Term

  • states that any sufficiently smooth function can be represented by a Taylor series plus a
    • The remainder term Rn(x)R_n(x) measures the error between the function and its Taylor polynomial approximation
    • As nn increases, the remainder term generally decreases, improving the approximation
  • Remainder term can be expressed in various forms, such as Lagrange or integral form
    • Lagrange form: Rn(x)=f(n+1)(c)(n+1)!(x−a)n+1R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}, where cc is between aa and xx
    • Integral form: Rn(x)=∫axf(n+1)(t)n!(x−t)ndtR_n(x) = \int_a^x \frac{f^{(n+1)}(t)}{n!}(x-t)^n dt
  • Analyzing the remainder term helps determine the accuracy of the Taylor polynomial approximation
    • Smaller remainder terms indicate better approximations
    • Remainder term can be used to bound the error and determine the number of terms needed for a desired accuracy

Key Terms to Review (20)

Absolute Convergence: Absolute convergence refers to the property of a series where the series formed by taking the absolute values of its terms converges. If a series converges absolutely, it guarantees that the original series also converges, which is a stronger condition than mere convergence. Understanding this concept is vital as it helps in applying various convergence tests and working with power series and Taylor series expansions effectively.
Approximating functions: Approximating functions involves creating simpler representations of complex functions using polynomials or series that closely mimic the behavior of the original function near a specific point. This technique is especially useful in calculus, allowing for easier computation and analysis, particularly through the use of Taylor and Maclaurin series which provide polynomial approximations of functions centered around a point.
Center of Expansion: The center of expansion refers to a specific point in the coordinate plane around which a function can be expressed as a Taylor or Maclaurin series. This point serves as the anchor for approximating the function using a polynomial formed from its derivatives at that location, providing a way to understand how the function behaves near that point. Essentially, the center of expansion allows for localized analysis and approximation of functions, making it crucial for applications involving calculus and analysis.
Coefficients: Coefficients are numerical factors that multiply variables in mathematical expressions, often seen in polynomial equations and series expansions. In the context of series expansions, coefficients play a crucial role in determining the value of each term in the series, allowing for approximations of functions using finite sums of powers of variables.
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 value at a point: The function value at a point refers to the output or result of a function when a specific input is provided. In the context of Taylor and Maclaurin series expansions, this concept is crucial as it helps in approximating complex functions using polynomial expressions centered around a particular point. Understanding how to calculate the function value at various points allows for deeper insights into the behavior of the function near those points, especially when dealing with series approximations.
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.
Linear Approximation: Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. This technique is based on the idea that a smooth function can be closely represented by a straight line in the immediate vicinity of that point, allowing for easier calculations and predictions.
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.
Mean Value Theorem: The Mean Value Theorem states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the derivative of the function is equal to the average rate of change over that interval. This theorem connects the concepts of continuity and differentiability, revealing crucial insights about the behavior of functions.
Nth derivative: The nth derivative of a function represents the result of differentiating that function n times. This concept extends the idea of the first and second derivatives, allowing us to analyze the behavior of functions more deeply, especially when studying polynomial behavior and approximating functions using series. The nth derivative is a key tool in understanding how functions change and is vital in the context of higher-order derivatives and series expansions.
Quadratic Approximation: Quadratic approximation is a method used to estimate the value of a function near a specific point by using the quadratic function derived from the function's Taylor series expansion. This technique captures the local behavior of the function by incorporating information about the function's value, first derivative, and second derivative at that point. It provides a simplified yet effective way to analyze complex functions in calculus and can be particularly useful for making predictions and solving optimization problems.
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.
Remainder Term: The remainder term is a concept that represents the difference between the actual value of a function and the approximation provided by its Taylor or Maclaurin series. It indicates how closely the series matches the function within a certain interval, helping to assess the accuracy of the approximation as more terms are added to the series.
Rolle's Theorem: Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval between two points, and the function has equal values at these two endpoints, then there exists at least one point in the open interval where the derivative of the function is zero. This theorem is crucial in understanding the behavior of functions and their derivatives, linking concepts like continuity and differentiability to critical points.
Taylor Polynomial: A Taylor polynomial is a polynomial approximation of a function that is derived from the function's derivatives at a single point. It provides a way to estimate the value of the function near that point by using its derivatives to create a series of terms. This concept connects to error estimation, different series expansions, and various practical applications where approximating functions is necessary.
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.
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.
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