Taylor Series in Calculus

Study Notes

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 ).

Description

This quiz explores the concept of Taylor series, including its definition, general formula, and convergence criteria. It also covers common Taylor series for exponential, sine, and cosine functions. Test your understanding of this essential topic in calculus!