Question

The necessary condition for the
Maclaurin expansion to be true for
function f(x) is​

Answer 1

Answer:

Any function which is infinitely differentiable at a point a has a Taylor series at that point. Whether or not the Taylor series converges at any point other than a is a different issue. But for the existence of a Taylor series all you need is the coefficients to exist, and these only require knowing the derivatives of the function at that point, so this is your sufficient condition. It is of course also necessary since if the function has a Taylor series, then the coefficients contain all higher derivatives at the point

Answer 2

• When a = 0, a Maclaurin series is obtained. When g(x) = f(x + a) is introduced, f(n)(a) = g(n)(0) is obtained, and the Maclaurin series for g at x = 0 corresponds to the Taylor series for f at x = a.

• A Taylor series exists at any point where a function is endlessly differentiable. It's a distinct question if the Taylor series converges at any point other than a.
• However, all that is required for the existence of a Taylor series is for the coefficients to exist, and knowing the derivatives of the function at that time is all that is required, therefore this is your sufficient condition.
• It's also required because, if the function has a Taylor series, the coefficients at the point contain all higher derivatives.