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Understanding Taylor and Maclaurin Series

May 11, 2025

Notes on Taylor and Maclaurin Series

Introduction to Series

  • Power Series: A series in the form ( C_n \cdot X^n ) from 0 to infinity.
  • The sum of a power series can represent a function.

Coefficients in Power Series

  • We want to solve for coefficients in series of the form ( C_n (X - A)^n ).
  • If we evaluate ( F(A) ), all terms except ( C_0 ) become zero, so ( F(A) = C_0 ).

Finding Coefficients

  1. First Derivative:

    • Take the derivative of the function, ( F' ).
    • ( C_0 ) goes to zero, leaving ( C_1 ).
    • If using the chain rule, the second coefficient is obtained as follows:
      • ( F'(A) = C_1 )

  2. Second Derivative:

    • Second derivative ( F'' ):
    • ( F''(A) = 2C_2 )
      • Thus, ( C_2 = \frac{F''(A)}{2} )

  3. Higher Derivatives:

    • The third derivative gives ( F'''(A) = 6C_3 )
      • Therefore, ( C_3 = \frac{F'''(A)}{6} )
    • This pattern continues:
      • General formula: ( C_n = \frac{F^{(n)}(A)}{n!} )
      • Where the ( n! ) is the factorial of n.

Taylor Series

  • The coefficients defined by: ( C_n = \frac{F^{(n)}(A)}{n!} )
  • The Taylor series is generated by the function: [ F(X) = \sum_{n=0}^{\infty} C_n (X - A)^n ]
  • Convergence occurs for ( |X - A| < \text{radius of convergence} )._

Special Case: Maclaurin Series

  • A Maclaurin series is a Taylor series at ( A = 0 ).
  • The series becomes: [ F(X) = F(0) + \frac{F'(0)}{1!}X + \frac{F''(0)}{2!}X^2 + \ldots ]

Example: Maclaurin Series for ( e^X )

  • The nth derivative of ( e^X ) is ( e^X ).
  • Evaluating at zero yields:
    • All derivatives equal 1 at zero.
  • Thus, the series becomes: [ \sum_{n=0}^{\infty} \frac{X^n}{n!} ]
  • This is the Maclaurin series for ( e^X )._

Radius of Convergence

  • Use the ratio test to find radius of convergence:
    • Set up as ( \frac{X^{n+1}}{(n+1)!} \cdot \frac{n!}{X^n} )
    • Results in ( \lim_{n \to \infty} \frac{|X|}{n+1} = 0 ) (converges for all ( X )).
  • Radius of convergence is infinite._

Conclusion

  • Taylor series have many applications; definitions are foundational.
  • Prepare to explore more topics after this foundational knowledge.

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