Taylor and Maclaurin series are powerful tools for approximating functions using polynomials. They allow us to represent complex functions as simpler polynomial expressions, making calculations and analysis much easier.

These series are especially useful when dealing with functions that are hard to evaluate directly. By using Taylor polynomials, we can get close approximations of function values and analyze their behavior near specific points.

Taylor and Maclaurin Series

Construction of Taylor polynomials

• Approximate functions near a specific point $a$ using Taylor polynomials
  • Denote the $n$th degree Taylor polynomial as $P_n(x)$
  • Express the general form of a Taylor polynomial as:
    • $P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$
• Recognize Maclaurin polynomials as a special case of Taylor polynomials where $a=0$
  • Write the general form of a Maclaurin polynomial as:
    • $P_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n$
• Construct a Taylor polynomial using the following steps:
  1. Select the degree $n$ of the polynomial
  2. Compute the derivatives $f'(a), f''(a), \ldots, f^{(n)}(a)$ at the point $a$
  3. Substitute the calculated derivatives and the point $a$ into the general form of the Taylor polynomial

Interpretation of Taylor's theorem

• State Taylor's theorem with remainder for a function $f(x)$ that is $n+1$ times differentiable on an interval containing $a$
  • Express the function as $f(x) = P_n(x) + R_n(x)$, where $R_n(x)$ is the remainder term
  • Define the remainder term as $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$, where $c$ is a value between $a$ and $x$
• Interpret the remainder term $R_n(x)$ as the error between the actual function value and the Taylor polynomial approximation
• Understand that increasing the degree $n$ of the Taylor polynomial generally decreases the remainder term
  • Conclude that higher-order Taylor polynomials provide more accurate approximations
• Apply Taylor polynomials to approximate functions in practical situations when:
  • Evaluating the function directly is challenging
  • The function is not explicitly known, but its derivatives at a point are available
  • A simpler approximation is required for numerical computations or analysis (e.g., for analytic functions like $e^x$, $\sin(x)$, $\cos(x)$)

Error analysis in Taylor series

• Identify the error in a Taylor series approximation as the remainder term $R_n(x)$
• Calculate the error using the following steps:
  1. Identify the degree $n$ of the Taylor polynomial
  2. Determine the $(n+1)$th derivative of the function $f^{(n+1)}(x)$
  3. Evaluate the $(n+1)$th derivative at a point $c$ between $a$ and $x$
  4. Substitute the values into the remainder term formula: $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$
• Analyze the error using the following properties:
  • Increasing the degree $n$ of the Taylor polynomial decreases the error
  • The error is smaller when the point $x$ is closer to the expansion point $a$ ($x=0.1$ vs. $x=1$)
  • The magnitude of the $(n+1)$th derivative of the function affects the error
• Determine bounds for the error in some cases using:
  • The Lagrange error bound: $|R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1}$, where $M$ is the maximum value of $|f^{(n+1)}(x)|$ on the interval between $a$ and $x$
  • The alternating series error bound (for alternating series): $|R_n(x)| \leq |a_{n+1}|$, where $a_{n+1}$ is the first neglected term in the series ($\sin(x)$, $\ln(1+x)$)

Convergence and Applications of Taylor Series

• Understand that Taylor series are infinite series representations of functions
• Determine the radius of convergence for a Taylor series
  • Identify the interval of convergence, which includes the radius and endpoints
• Recognize that Taylor series converge to the original function within their interval of convergence
• Apply Taylor series to various mathematical and practical problems, such as:
  • Approximating complex functions
  • Solving differential equations
  • Analyzing the behavior of functions near specific points