Taylor Series

In mathematics, the Taylor Series is one of the most powerful tools for approximating functions.

• Study of Taylor series is largely about taking non-polynomial functions and finding polynomials that approximate them near some input
• The motive is that polynomials are much easier to deal with than other functions:
  • easier to compute
  • easier to take derivatives
  • easier to integrate
• Idea: Leverage derivative information at a point to approximate output near that point
• "Taylor polynomials" are the result of applying this principle

General approach:

• Compute derivative, 2nd derivative etc. of f(x)
• Evaluate each one of them at x₀ (point to approximate around)
• Put in the value of the n-th derivative as coefficients for the x_n term divided by the factorial n!
• for approximation around 0: \[ P(x) = f(0) + \frac{d}{dx}f(0) \frac{x^{1}}{1!} + \frac{d^{2}}{dx}f(0) \frac{x^{2}}{2!} + ... \]

• general formula for a point a:

  \[ P(x) = f(a) + \frac{df}{dx}(a) \frac{(x-a)^{1}}{1!} + \frac{d^{2f}}{dx}(a) \frac{(x-a)^{2}}{2!} + ... \]

• Radius of convergence: The maximum distance between the input you're approximating near, and points where the outputs of these polynomials actually do converge. I.e. a approximation point outside this radius will not create a perfect approximation of the function.