Taylor and Maclaurin Series - Calculus Volume 2

Learning Objectives

• Describe the procedure for finding a Taylor polynomial of a given order for a function.
• Explain the meaning and significance of Taylor's theorem with remainder.
• Estimate the remainder for a Taylor series approximation of a given function.

Overview of Taylor/Maclaurin Series

• Power Series Representation: Consider a function ( f ) that has a power series representation at ( x = a ): [ \sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \ldots ]
• Coefficients are determined by derivatives of ( f ) at ( a ):
  • ( c_0 = f(a) )
  • ( c_1 = f'(a) )
  • ( c_2 = \frac{f''(a)}{2!} )
  • ( c_3 = \frac{f'''(a)}{3!} )
• Taylor Series: If ( f ) has derivatives of all orders at ( x = a ), the Taylor series is: [ \sum_{n=0}^{\infty} \frac{f^n(a)}{n!} (x-a)^n ]
• Maclaurin Series: A Taylor series centered at ( a = 0 ).

Uniqueness of Taylor Series

• If ( f ) has a power series at ( a ) that converges, it is unique and must be the Taylor series for ( f ).

Taylor Polynomials

• Definition: The ( n )-th partial sum of the Taylor series is the ( n )-th Taylor polynomial: [ p_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots + \frac{f^n(a)}{n!}(x-a)^n ]
• Maclaurin polynomials are Taylor polynomials centered at zero.

Taylor's Theorem with Remainder

• Remainder: ( R_n(x) = f(x) - p_n(x) )
• Theorem states ( R_n(x) ) for a function ( f ) that is ( n+1 ) times differentiable on an interval ( I ) is: [ R_n(x) = \frac{f^{n+1}(c)}{(n+1)!}(x-a)^{n+1} ]
• Provides a bound for the remainder and estimates the error in approximation.

Examples and Exercises

• Examples demonstrate finding Taylor and Maclaurin polynomials and estimating errors.

Example 6.11

• Finding Taylor polynomials for ( f(x) = \ln x ) at ( x=1 ).

Example 6.12

• Finding Maclaurin polynomials for functions like ( e^x ), ( \sin x ), ( \cos x ).

Example 6.13

• Using Taylor polynomials to estimate values and bounds on errors.

Example 6.14

• Approximating ( \sin x ) using Maclaurin polynomials.

Representing Functions with Taylor and Maclaurin Series

• Discusses convergence of Taylor series and finding intervals of convergence.

Example 6.15

• Finding Taylor series for ( f(x) = \frac{1}{x} ) at ( x=1 ).

Theorem 6.8: Convergence of Taylor Series

• A Taylor series converges to ( f(x) ) if and only if the remainder ( R_n(x) ) approaches zero.

Exercises

• Various exercises to apply concepts, such as finding Taylor series, verifying remainder estimates, and using Taylor polynomials for approximation.

Student Project: Proving that ( e ) is Irrational

• Uses Maclaurin polynomials to demonstrate that ( e ) cannot be a rational number.

These notes provide a comprehensive summary of Taylor and Maclaurin series, their properties, computation, and applications. They also offer practice exercises to understand the concepts better.