Taylor Series and Approximation

1. Taylor Polynomial

At point a:

f(x)=sum_{k=0}^n f^(k)(a)/k! * (x-a)^k + R_n(x).

R_n is remainder (error term).

2. Common Expansions at 0

• e^x = 1 + x + x^2/2! + x^3/3! + ...
• sin x = x - x^3/3! + x^5/5! - ...
• cos x = 1 - x^2/2! + x^4/4! - ...
• ln(1+x) = x - x^2/2 + x^3/3 - ... for |x|<1

3. Error Bounds

Lagrange remainder provides bound using (n+1)-st derivative.

4. Why CS Uses This Constantly

• local model in optimization
• numerical function implementations
• complexity approximations for log/exp near operating point
• control and robotics linearization

5. Worked Example

Approximate e^0.1 with quadratic polynomial: 1 + 0.1 + 0.1^2/2 = 1.105. True value ~1.10517, very close.

6. Asymptotic Approximation Idea

Keep dominant terms when variable is small/large. This supports both numerical and algorithmic reasoning.

Exercises

1. Approximate sin(0.2) using 3rd-order polynomial.
2. Bound approximation error using next derivative term.
3. Use series to derive lim_{x->0} (e^x-1)/x.
4. Explain relation of second-order Taylor model to Newton updates.