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> Due to Taylor’s theorem, any differentiable real-valued function may be approximated by polynomials so Cauchy’s method is applicable to any function that is d
by mehrdadn 7y ago
> Due to Taylor’s theorem, any differentiable real-valued function may be approximated by polynomials so Cauchy’s method is applicable to any function that is differentiable.
Sorry, what? Pretty sure this is not Taylor's theorem (and it's false)?
- g82918 7y agoShould be Stone-Wierstrauss, and it is true on compact regions only(and requires a few other bits).
- dan-robertson 7y agoWell Taylor’s theorem is that any sufficiently differentiable function can be approximated by a particular polynomial plus a particular error. It just turns out that the error term doesn’t need to be very well behaved (eg consider the Taylor series at 0 for f(x)=exp(-1/x^2) ). I think instead this is an application of the fundamental theorem of applied maths which states, approximately, that, in applied mathematics: - all Taylor series converge - all functions are piecewise smooth - all sums and integrals can be transposed - if the solution must be x if it exists, then the solution exists (and is x) - if it looks right then it is