Lecture on Taylor and McLaurin Series

Overview

• Purpose: Represent functions with power series.
• Focus: Taylor series and McLaurin series.

Power Series Representation

• General Form:
  • If a function ( f ) has a power series representation at a point ( C ), then: [ f(x) = \sum_{n=0}^{\infty} a_n (x-c)^n ]
  • ( a_n ) are constants, ( x-c ) is raised to powers from 0 to ( n ).
  • If ( C = 0 ), it's still a power series._

Derivatives and Power Series

• If a function can be represented by a power series, derivatives of all orders must exist.
• Nth Derivative:
  • The nth derivative exists and can be found by repeatedly applying the derivative rules.
  • Coefficients ( a_n ) relate to derivatives as: [ a_n = \frac{f^{(n)}(c)}{n!} ]

Taylor Series

• Definition: (Centered at ( C ))
  • [ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x-c)^n ]
• Steps to Find Taylor Series: (When C = 0, it's a McLaurin series)
  1. Find several derivatives of the function.
  2. Evaluate them at ( x = c ).
  3. Find a pattern and express the nth derivative.
  4. Use the pattern to assemble the series.
  5. Determine the interval of convergence using the Ratio Test._

McLaurin Series

• Special case of Taylor series where ( c = 0 ).
• Series Representation:
  • [ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n ]_

Example Calculations

• Example 1: ( f(x) = e^x )

  • Derivatives repeat: ( f^{(n)}(x) = e^x ).
  • McLaurin Series: ( \sum_{n=0}^{\infty} \frac{x^n}{n!} )
  • Converges for all real ( x ).

• Example 2: ( f(x) = \ln(x) )

  • Derivatives involve decreasing power of ( x ).
  • Taylor series might start at a non-zero derivative if the pattern starts later.
  • Interval of convergence can be affected by which derivatives vanish._

Binomial Series

• General Form: ( (1 + x)^k )
  • Derived using the formula for derivatives and expanded using Taylor series.
  • Specific cases depending on ( k )'s values.
• Convergence:
  • If ( k ) is a positive integer, converges for all ( x ).
  • Otherwise, convergence depends on the value of ( k ).

Common Series to Remember

1. Exponential Function ( e^x )
   • ( \sum_{n=0}^{\infty} \frac{x^n}{n!} )
2. Sine Function ( \sin(x) )
   • ( \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} )
3. Cosine Function ( \cos(x) )
   • ( \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} )
4. Natural Logarithm ( \ln(1+x) )
   • ( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} )
5. Binomial Series ( (1+x)^k )
   • ( \sum_{n=0}^{\infty} \binom{k}{n} x^n )_

Using Known Series for Other Functions

• Manipulate functions to fit the form of known series using algebraic manipulations.
• Example: Express ( \frac{1}{1+x} ) by transforming variables and using geometric series.

Integration of Series

• Can integrate a power series term-by-term.
• Example: Integrating ( e^{-x^2} ) series.

Closing Thoughts

• Taylor and McLaurin series are powerful tools for approximating functions using polynomials.
• Important to understand the convergence properties to ensure series accurately represent functions over intended intervals.