Lecture Notes on Taylor Series

Importance of Taylor Series

• Taylor series are crucial in mathematics, physics, and engineering.
• They approximate functions, making them easier to handle in calculations.

Key Example: Pendulum and Cosine Function

• Cosine function complicates calculations for the potential energy of a pendulum.
• Approximation of cosine: ( \cos(\theta) \approx 1 - \frac{\theta^2}{2} ) simplifies the problem.
• Valid for small angles near 0 (graph comparison shows similarity).

Constructing a Quadratic Approximation

• Aim: Create a polynomial approximation near ( x = 0 ).
• General form: ( c_0 + c_1 x + c_2 x^2 )
• Criteria for matching ( \cos(x) ):
  • Match value: Set ( c_0 = 1 ) (since ( \cos(0) = 1 )).
  • Match first derivative: Set ( c_1 = 0 ) (flat tangent at ( x = 0 )).
  • Match second derivative: Set ( c_2 = -\frac{1}{2} ) (to ensure curvature matches)

Resulting Quadratic Approximation

• The approximation becomes: ( 1 - \frac{1}{2} x^2 )
• Estimation example: ( \cos(0.1) \approx 0.995 ) (accurate).

Higher Order Terms

• Adding terms improves approximation:
  • Cubic term: ( c_3 x^3 ) leads to ( c_3 = 0 ).
  • Quartic term: ( c_4 x^4 ) leads to ( c_4 = \frac{1}{24} ).
• Quartic approximation: ( 1 - \frac{1}{2} x^2 + \frac{1}{24} x^4 )

Key Observations

• Factorial terms arise naturally; must divide derivatives by factorial for coefficients.
• Adding new terms doesn't disrupt previous terms due to evaluating at ( x = 0 ).
• For approximating at a different point (e.g., ( x = \pi )), use powers of ( x - a ).

Philosophical Insight

• Derivative values give insight into function behavior at a point.
• Taylor polynomials use derivative information to create polynomial approximations.
• General formula for Taylor polynomial:
  • Coefficient of ( x^n ): ( \frac{f^{(n)}(0)}{n!} )

Example: Taylor Series for ( e^x )

• All derivatives of ( e^x ) at ( x = 0 ) equal 1.
• Taylor polynomial: ( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots )

Geometric Understanding of Second Order Term

• Relates to the Fundamental Theorem of Calculus.
• Area function approximation looks at height and slope of the graph.
• Triangle area formulation relates closely to Taylor polynomial terms.

Taylor Series vs. Taylor Polynomials

• Taylor series sums up infinitely many terms; convergence depends on function characteristics.
• Example: ( e^x ): Taylor series converges at all points.
• Example: Natural log ( \ln(x) ): Converges only in specific ranges (diverges outside).

Radius of Convergence

• Defined as the distance from the point of approximation where the series converges.

Conclusion and Next Steps

• Taylor series translate derivative information into approximations.
• More topics on Taylor series and calculus await further exploration.