Taylor Series in Calculus

Taylor Series

• Definition: A Taylor series is an infinite series that represents a function as a sum of terms calculated from the values of its derivatives at a single point.

• General Formula:

  • For a function ( f(x) ) that is infinitely differentiable at a point ( a ): [ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots ]
  • Or in summation notation: [ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n ]

• Convergence:

  • A Taylor series may converge to ( f(x) ) within a certain interval around ( a ).
  • The radius of convergence can be determined using the Ratio Test or the Root Test.

• Common Taylor Series:

  • Exponential Function: [ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} ]
  • Sine Function: [ \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} ]
  • Cosine Function: [ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} ]
  • Natural Logarithm (around ( x = 1 )): [ \ln(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-1)^n}{n} ]

• Applications:

  • Approximation of functions: Taylor series can be used to approximate functions near the point ( a ).
  • Numerical methods: Used in algorithms for root-finding and optimization.
  • Analysis of functions: Helps in understanding the behavior of functions near specific points.

• Maclaurin Series: A special case of the Taylor series centered at ( a = 0 ): [ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n ]

• Error in Approximation:

  • The remainder term ( R_n(x) ) indicates the error of truncating the series after ( n ) terms: [ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} \quad \text{for some } c \text{ between } a \text{ and } x. ]

• Key Considerations:

  • Not all functions can be represented by a Taylor series.
  • Functions may have finite derivatives but still not be representable as Taylor series (e.g., ( f(x) = e^{-1/x^2} ) for ( x \neq 0 )).

Taylor Series Overview

• A Taylor series represents a function as an infinite sum of terms derived from derivatives at a single point.
• The general formula for a Taylor series for a function ( f(x) ) at point ( a ) involves terms of ( f(a) ) and its derivatives: [ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots ]
• In summation notation, it can be expressed as: [ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n ]

Convergence of Taylor Series

• The convergence of a Taylor series to ( f(x) ) occurs within a specific interval surrounding ( a ).
• The radius of convergence can be assessed via the Ratio Test or Root Test.

Common Taylor Series Examples

• Exponential Function: [ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} ]
• Sine Function: [ \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} ]
• Cosine Function: [ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} ]
• Natural Logarithm (around ( x = 1 )): [ \ln(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-1)^n}{n} ]

Applications of Taylor Series

• Function Approximation: Useful for approximating functions near point ( a ).
• Numerical Methods: Implemented in algorithms for tasks like root-finding and optimization.
• Function Analysis: Aids in exploring functions' behavior close to specific points.
• Maclaurin Series: A special Taylor series where the expansion is centered at ( a = 0 ): [ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n ]

Error in Approximation

• The remainder term ( R_n(x) ) quantifies the truncation error after ( n ) terms: [ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} \quad \text{for some } c \text{ between } a \text{ and } x. ]

Key Considerations

• Not every function can be represented by a Taylor series.
• Some functions may possess finite derivatives but are not expressible as Taylor series, such as ( f(x) = e^{-1/x^2} ) for ( x \neq 0 ).