Lecture Notes on Taylor Series

Introduction to Taylor Series

• Taylor series are crucial for approximating functions in math, physics, and engineering.
• Initial realization of their importance often occurs in physics contexts.

Example: Pendulum and Cosine Function

• The potential energy of a pendulum involves the cosine function, which complicates relationships with other oscillating phenomena.
• Approximation: Cosine of theta can be expressed as:
  • cos(θ) ≈ 1 - (θ²)/2 for small angles near 0.
• Graphing both functions shows they are close for small angles but finding this approximation requires understanding Taylor series.

Taylor Series Concept

• Taylor series involve taking non-polynomial functions and finding polynomial approximations near a specific input.
• Polynomials are easier to compute, differentiate, and integrate.

Constructing a Quadratic Approximation for cos(x)

1. Matching the Value at x = 0:
   • cos(0) = 1 → set c0 = 1.
2. Matching the Derivative at x = 0:
   • The derivative of cos(x) is -sin(x) → equals 0 at x = 0.
   • Set c1 = 0 for a flat tangent line at x = 0.
3. Matching the Second Derivative at x = 0:
   • Second derivative of cos(x) is -cos(x) → equals -1 at x = 0.
   • Set c2 = -1/2.

• Final Quadratic Approximation:
  • P(x) = 1 - (1/2)x².
• This approximation is close to the true value of cos(0.1).

Higher-Order Approximations

• Adding higher-order terms (like c3 * x³) requires matching third derivatives, leading to c3 = 0.
• Including a fourth-order term (c4 * x⁴) results in:
  • Set c4 = 1/24.
• Fourth-Order Polynomial Approximation:
  • P(x) = 1 - (1/2)x² + (1/24)x⁴.

Observations During Approximation

• Factorials arise naturally when working with derivatives.
• Each coefficient of the polynomial affects specific derivatives of the function.

Generalizing Taylor Polynomials

• For a general function f(x) at x = a:
  • P(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)²/2! + ... + f⁽ⁿ⁾(a)(x-a)ⁿ/n!
• The nth derivative at point a informs the polynomial's nth coefficient.

Special Case: Taylor Series for e^x

• All derivatives of e^x are equal to e^x → at x = 0, all derivatives equal 1.
• Taylor polynomial is:
  • P(x) = 1 + x + x²/2! + x³/3! + ...

Geometric Interpretation with Area Function

• Relate Taylor polynomials to the area under a curve.
• Use the fundamental theorem of calculus to visualize changes in area, leading to the second-order term in Taylor polynomials.

Infinite Series and Convergence

• Taylor series are constructed by summing infinitely many terms.
• Convergence: Series approaches a specific value as more terms are added.
• Example: Taylor series for e^x converges to e^1 when x = 1.
• Divergence: Series may fail to converge outside a certain radius, such as the Taylor series for ln(x) around x = 1.

Conclusion

• Taylor series translate derivative information at a single point into approximation information around that point.
• Understanding Taylor series provides a foundation for further studies in calculus and related fields.
• Future series will explore probability.