Applications of Taylor Series

In this lecture, we explore three applications of Taylor series to solve problems that were challenging or impossible to solve using only Calculus 1 and Calculus 2 methods.

Application 1: Integration

• Problem: Integral of ( e^{-x^2} , dx )
  • Important in probability and statistics (normal distribution).
  • Traditional methods (u-substitution, integration by parts) do not work.
• Solution with Taylor Series:
  • Use the Taylor series for ( e^x ): [ \sum_{n=0}^{\infty} \frac{x^n}{n!} ]
  • Substitute (-x^2) for (x), and integrate term by term:
    • ( \sum (-1)^n \frac{x^{2n+1}}{n!(2n+1)} + C )
  • Allows for approximation using finite terms.

Application 2: Limits

• Problem: Evaluate limits involving indeterminate forms (( \frac{0}{0} )).
  • Example: ( \frac{x^2(e^x - \cos x)}{-x^2/2} ) as ( x \to 0 ).
• Solution with Taylor Series:
  • Expand ( e^x ) and ( \cos x ) as series:
    • ( e^x \approx 1 + x + \frac{x^2}{2} + \cdots )
    • ( \cos x \approx 1 - \frac{x^2}{2} + \cdots )
  • Simplify numerator and denominator:
    • Cancel higher-order terms.
    • Focus on dominating terms as ( x \to 0 ).
  • Result: Limit simplifies to (-2).

Application 3: Series Evaluation

• Problem: Evaluate specific series to determine convergence and sum.
  • Example: Series similar to ( e^x ) with all ones (( x = 1 )).
• Solution with Taylor Series:
  • Recognize series as Taylor series for ( e^x ):
    • ( 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots = e )
  • Variants with alternating signs can be recognized as ( e^{-x} ).
    • Example: Plugging ( x = -1) results in ( e^{-1} = \frac{1}{e} ).

In summary, Taylor series allow us to:

• Solve integrals that don't have elementary antiderivatives.
• Evaluate limits more effectively.
• Determine the value to which a series converges.