Fix n. Let a real function f be approximated by the first n terms of
its Maclaurin expansion:

f(x) \approx F(x) = f(0) + f'(0)*x + f''(0)*x^2/2 + ... +
f^{(n-1)}*x^{n-1}/n! .

Let f also be approximated by an n-term polynomial:

f(x) \approx P(x) = a_0 + a_1*x + a_2*x^2/2 + .... + a_{n-1}*x^{n-1}/n!.

To approximate f with P, will the "best" choice of a_k always be
f^{(k)}/k! ?

I realize that there are different kinds of closeness of approximation
which may lead to different choices of a_k. Let's say we wish to
minimize

\int_a^b |f(x) - P(x)| dx

for some interval (a, b) in f's domain.

Suppose f(x) = exp(-1/x) for x > 0, f(x) = 0 otherwise. Then f is C^oo on
R, and all MacLaurin polynomials of f are 0. So clearly another polynomial
will do better in almost any kind of approximation, including the integral
approximation.

However, given any smooth f and any polynomial P of degree n that is not
the MP of degree n, there exists b > 0 such that that the integral
approximation of f over (0,b) with MP_n is better than that with P.

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I will move this Thread to; ' Math themes '


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saucer