Calc 2 - 10/27

Key Concepts of Taylor and Maclaurin Series

• Power Series Representation: A function ( f(x) ) can be expressed as a power series around a point ( a ).

• Nth Derivative Notation: ( f^{(n)} ) denotes the nth derivative of ( f ) with respect to ( x ).

• Factorial Definitions:

  • ( n! = n \times (n-1) \times (n-2) \times \ldots imes 1 )
  • ( 0! = 1 ) and ( 1! = 1 )

Coefficient Evaluation

• The coefficient ( c_n ) can be found using:

  • ( c_n = \frac{f^{(n)}(a)}{n!} )

• Evaluating function and derivatives at point ( a ) gives important coefficients:

  • ( f(a) = 0!c_0 )
  • ( f'(a) = 1!c_1 )
  • ( f''(a) = 2!c_2 )
  • ( f^{(n)}(a) = n!c_n )

Taylor Series Definition

• If ( f ) has derivatives of all orders at point ( a ), then:
  • ( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n )

Maclaurin Series Special Case

• When the center ( a = 0 ):
  • ( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n )

Examples and Applications

• Example for Maclaurin Series:

  • For ( f(x) = e^x ), its Maclaurin series is:
  • ( f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} )

• Finding Interval of Convergence:

  • Typically determined using the Ratio Test;
  • If series converges, it converges for all ( x ).

Exercise Directions

• To find Taylor or Maclaurin series, identify the required derivatives,
  • Substitute into the series formula and simplify to desired form.
• Confirm the center for Maclaurin series is always zero unless specified otherwise.