In which we try to approximate functions by polynomials.
- Theorem 28 (Taylor‘s theorem with Lagrange‘s form of the remainder) Let be a function such that and its first derivatives are continuous on and is times differentiable on . Then there is some such that . We proved this by applying Rolle‘s theorem times to a suitable auxiliary function.
- Theorem 29 (Taylor’s theorem with Cauchy‘s form of the remainder) Let be a function such that and its first derivatives are continuous on and is times differentiable on . Then there is some such that . We proved this by applying the Mean value theorem to a suitable auxiliary function.
Understanding today’s lecture
A good way to understand the proofs would be to work through them for particular small values of . You could try experimenting with other functions that you know about (we’ll meet standard functions such as exponentials and trig functions formally in lectures during the next couple of weeks).
Any introductory analysis textbook is likely to have a section on Taylor’s theorem with various forms of the remainder.
Preparation for Lecture 14
Can you finish our analysis of the binomial example ( where is rational)?
Remind yourself of the definition of differentiability for functions from to . At which points is the complex conjugate function differentiable?
If I give you a convergent series (where ), what can you say about for ? If I give you a series that does not converge, what can you say about for ?