Power Series

Taylor Series

The Taylor series is a powerful tool in calculus for representing functions as infinite sums of their derivatives evaluated at a single point. It is a specific type of power series that approximates functions near a given point, often used for analysis, computation, and solving differential equations.

• Definition: The Taylor series of a function f(x) centered at a is given by:

• Key Terms:

  • Power series: An infinite sum of the form .

  • Maclaurin series: A Taylor series centered at a = 0.

• Convergence: The Taylor series converges to f(x) within the radius of convergence, which depends on the function and the point a.

• Applications:

  • Approximating functions near a point

  • Solving differential equations

  • Calculating limits and integrals

Examples of Taylor Series

• Maclaurin Series for :

• Maclaurin Series for :

• Maclaurin Series for :

Properties and Comparison

• Accuracy: The more terms included, the closer the approximation to the actual function within the radius of convergence.

• Remainder (Error): The error in approximating f(x) by its Taylor polynomial of degree n is given by: where c is between a and x.

Table: Common Taylor Series (Maclaurin Series)

Example: To approximate using the first three terms of its Maclaurin series:

Additional info: The Taylor series is a central concept in Chapter 11 (Power Series) and is foundational for advanced calculus topics, including numerical analysis and mathematical modeling.