Calc - 10/31

Power Series and Functions

Overview

  • Discusses the operations applicable to functions represented by power series expansions when the functions share the same interval of convergence.

First Statement: Addition and Subtraction of Functions

  • Functions: Consider two functions, ( f(x) ) and ( g(x) ).
  • Representation: Both can be expressed as power series:
    • Power series expansion requires both functions to have the same interval of convergence.
  • Combining Functions:
    • For addition: ( f + g ) can be represented as a power series where the coefficients are the sum of the original coefficients:
    • ( an + bn ) for corresponding terms.
    • For subtraction: ( f - g ) can also be represented similarly.

Second Statement: Multiplication of Functions

  • Functions: Again, consider functions ( f(x) ) and ( g(x) ).
  • Multiplication of Functions:
    • The product ( f imes g ) can be expanded into a power series with coefficients determined through multiplication of the original coefficients:
    • For example,
      • The coefficient of the constant term is ( a0 b0 ).
      • The coefficient of ( x ) is ( a0 b1 + a1 b0 ).
    • Conceptually similar to multiplying polynomials, but involves infinite terms.
  • Important Note: Practical application is not explicitly needed at this stage.

Third Statement: Division of Functions

  • Division Condition: Division of two power series is only valid provided that the divisor is not zero.
  • Interval of Convergence for Division:
    • The interval of convergence for ( \frac{f}{g} ) will be at least as large as that of ( f ).
    • Determining the exact interval of convergence after division is complex.

Application: Limit Evaluation Using Maclaurin Series

  • Problem Statement: Evaluate the limit ( \lim_{x \to 0} \frac{\cos(x) - 1 + \frac{x^2}{2}}{x^4} )
  • Hint: Use the Maclaurin series for ( \cos(x) ).
  • Maclaurin Series for Cosine:
    • ( \cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots )

Steps to Solve the Limit Problem

  1. Expand: Replace ( \cos(x) ) in the expression with its Maclaurin series.
  2. Cancellation:
    • Simplifying: Recognize terms such as ( -1 + \frac{x^2}{2} ) will cancel the first terms of the series.
  3. Limit Computation: After simplification, take the limit as ( x ) approaches zero.
  4. End Result: The remaining terms will guide toward the final result, helping in finding the limit.

Discussion on Derivatives

  • A question may arise regarding the derivative application: Yes, L'Hôpital's Rule can provide an alternative approach to such limits, particularly when faced with ( \frac{0}{0} ) indeterminate forms.
  • Different methods can yield the same results by substituting function definitions.

Taylor Polynomials and Remainders

Introduction to Taylor Series
  • Functions can be expressed via Taylor or Maclaurin series, representing their expansions around a specific center.
Definition of Taylor Polynomials
  1. Important Definitions:
    • Taylor Series: ( T_n(x) = \frac{f^{(0)}(a)}{0!}(x-a)^0 + \frac{f^{(1)}(a)}{1!}(x-a)^1 + \ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n )
    • This polynomial is an approximation that may lack infinite series continuity due to its finite nature.
  2. Degree of Polynomial:
    • This is referred to as the ( n )-th degree Taylor polynomial.
    • In the case when ( a = 0 ), it’s termed the Maclaurin polynomial.
Remainder of Taylor Series
  1. Definition of Remainder:
    • The error or remainder between the actual function and its Taylor approximation is represented as:
      • ( Rn(x) = f(x) - Tn(x) )
  2. Upper Bound on Error:
    • The error can be estimated by:
      • ( |R_n(x)| \leq \frac{M}{(n+1)!}|x - a|^{(n+1)} )
      • Where ( M ) is the maximum absolute value of the ( (n+1) )-th derivative on the interval considered.
Example Exercise
  • Find the third-degree Maclaurin polynomial for ( f(x) = e^x ):
    1. Derive the polynomial based on Taylor series expansion and identify finite terms.
    2. Calculate the approximation of ( e^1 ) using the derived polynomial.
    3. Obtain the estimate for the error in the approximation using the remainder formula.

Further Questioning

  • Q: Do you need to derive common functions like ( \cos(x), e^x, \sin(x) ) by yourself for exams?

    • A: Yes, it would be expected to derive these functions without pre-given series equations.

  • Discuss the implications of choosing larger numbers for the remainder bounds and its effect on the accuracy of the estimates.