Taylor series are powerful tools for representing functions as infinite sums of terms. They allow us to approximate complex functions using polynomials, making calculations easier. This concept builds on earlier calculus topics, extending our ability to analyze and work with functions.

Understanding Taylor series helps us tackle problems in advanced calculus and beyond. We can use them to estimate function values, solve differential equations, and even model real-world phenomena. It's a key technique that opens doors to more sophisticated mathematical analysis.

Taylor Series: Concept and Derivation

Definition and Derivation

• Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point
• Derived by considering the nth-degree Taylor polynomial and taking the limit as n approaches infinity
• Generalizes the Maclaurin series, which is a Taylor series expansion around the point x=0
• The Taylor series expansion of a function $f(x)$ around a point $a$ is given by the formula: $f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$

Coefficients and Point of Expansion

• The coefficients of the Taylor series are determined by the values of the function and its derivatives at the point of expansion
• The point of expansion, usually denoted as $a$ or $x_0$, is the point around which the series is expanded
• The Maclaurin series is a special case of the Taylor series where the point of expansion is $x=0$
• Examples of points of expansion:
  • Maclaurin series: $x=0$
  • Taylor series of $f(x)$ around $x=1$: $a=1$

Taylor Series Representation of Functions

Procedure for Finding Taylor Series

• To find the Taylor series of a function, first identify the point of expansion (usually denoted as $a$ or $x_0$)
• Calculate the function's value and the values of its derivatives at the point of expansion
• Substitute these values into the general formula for the Taylor series expansion
• Simplify the resulting expression to obtain the Taylor series representation of the function
• Example: To find the Taylor series of $f(x)=e^x$ around $x=0$, calculate $f(0), f'(0), f''(0), ...$ and substitute into the formula

Common Functions and Their Taylor Series

• Many common functions, such as exponential, trigonometric, and logarithmic functions, have well-known Taylor series expansions
• Examples of Taylor series for common functions:
  • $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$
  • $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$
  • $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ...$
• Knowing these expansions can help in finding Taylor series for more complex functions

Taylor Polynomials for Approximation

Definition and Properties

• Taylor polynomials are finite-degree polynomials that approximate a function near a given point
• The nth-degree Taylor polynomial is obtained by truncating the Taylor series after the nth term
• As the degree of the Taylor polynomial increases, the approximation becomes more accurate near the point of expansion
• Example: The 2nd-degree Taylor polynomial of $e^x$ around $x=0$ is $1 + x + \frac{x^2}{2!}$

Approximating Functions with Taylor Polynomials

• To approximate a function using a Taylor polynomial, substitute the desired value of $x$ into the polynomial and calculate the result
• Taylor polynomials are useful for approximating functions that are difficult to evaluate directly or for simplifying complex expressions
• Example: To approximate $e^{0.1}$ using a 3rd-degree Taylor polynomial, substitute $x=0.1$ into $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}$

Accuracy and Error Bounds of Taylor Series Approximations

Factors Affecting Accuracy

• The accuracy of a Taylor series approximation depends on the number of terms used and the distance from the point of expansion
• As more terms are included in the approximation, the accuracy generally improves
• The closer the value of $x$ is to the point of expansion, the more accurate the approximation tends to be
• Example: A 5th-degree Taylor polynomial of $\sin(x)$ around $x=0$ is more accurate for $x=0.1$ than for $x=1$

Error Bounds and Convergence

• The remainder term in the Taylor series represents the error between the function and its Taylor polynomial approximation
• The Lagrange remainder theorem provides an upper bound for the error in a Taylor polynomial approximation
• The error bound can be used to determine the number of terms needed to achieve a desired level of accuracy
• In some cases, the Taylor series may not converge for certain values of $x$, limiting the range of valid approximations
• The radius of convergence of a Taylor series determines the interval on which the series converges to the function
• Example: The Taylor series for $\ln(1+x)$ has a radius of convergence of 1, meaning it converges for $|x| < 1$