Taylor epxansion of an extended odd function x^2

Question

Let's say I have a function f given by:

f(x)=x^2 if x>0 otherwise -x^2

and the Taylor expansion given by:

Tf(x)=∑a_i x^i

So I assume the Taylor expansion of the function will contain only odd powers. How to compute it?

Answer 1

The Taylor expansion to the zeroth order is $0$.

The Taylor expansion to the first order is $0+0\cdot x$.

The Taylor expansion to the second order does not exist because of $f''(0)$.

Answer 2

Taylor expansion is extension of intermediate theorem, so it’s limited to (x, x+h), so I guess it’s inherently single side? It’s double side only when $f^{(n)}(x+0)= f^{(n)}(x-0)$, which in your example is not true.

Beside Taylor expansion is approximation of function by powers. So in your examples you get the expansion just as itself.

So you will get the Taylor expansion like $f(x+0)=x^2, f(x-0)=-x^2$, which involves powers of even order but simply negative of each other on two sides.