Calculus II - Taylor Series

Introduction

• Previous approach to power series: Relating functions to ( \frac{1}{1-x} ).
• Need for a general method for power series representation.

Assumptions for Power Series Representation

1. Function ( f(x) ) has a power series about ( x = a ):

   [ f(x) = \sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1 (x-a) + c_2 (x-a)^2 + \dots ]

2. ( f(x) ) has derivatives of all orders._

Determining Coefficients ( c_n )

• Evaluate at ( x = a ):

  [ f(a) = c_0 ]

  • All terms except first are zero.

• Taking derivatives:

  • First derivative at ( x = a ):

    [ f'(a) = c_1 ]

  • Second derivative:

    [ f''(a) = 2c_2 \Rightarrow c_2 = \frac{f''(a)}{2!} ]

  • Third derivative:

    [ f'''(a) = 3!c_3 \Rightarrow c_3 = \frac{f'''(a)}{3!} ]

  • General formula:

    [ c_n = \frac{f^{(n)}(a)}{n!} ]

Taylor Series

• For a function ( f(x) ) about ( x = a ):

  [ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n ]_

Maclaurin Series

• Special case of Taylor Series about ( x = 0 ):

  [ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n ]_

Existence of Taylor Series

• Define nth degree Taylor polynomial:

  [ T_n(x) = \sum_{i=0}^{n} \frac{f^{(i)}(a)}{i!} (x-a)^i ]

• Remainder or error between function and Taylor polynomial:

  [ R_n(x) = f(x) - T_n(x) ]

• Theorem:

  • If ( \lim_{n \to \infty} R_n(x) = 0 ) for ( |x-a| < R ), the Taylor series converges to ( f(x) ).

Examples

1. Taylor Series for ( f(x) = e^x ) about ( x = 0 ).
2. Taylor Series for ( f(x) = x^4 e^{3x^2} ) about ( x = 0 ).
3. Taylor Series for ( f(x) = cos(x) ) about ( x = 0 ).
4. Taylor Series for ( f(x) = ln(x) ) about ( x = 2 ).
5. Various other examples including cases not around ( x = 0 ).

Important Taylor Series Formulas

1. ( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} )
2. ( \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} )
3. ( \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} )_

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• Page last modified on 11/16/2022.