Before I start, I apologize if I use wrong mathematical terms as I'm studying maths in French. For example for us Taylor's Expansions are called Limited Development, not to be confused with Taylor's Theorem or Formula, as the former looks like this and doesn't require f to be n times differentiable, unlike Taylor's formula which does and looks like this (However, the latter if it exists equals the former, else, we can say the expansion exists).
Now, as the pictures showed, Taylor's expansion (Limited Development) consists of a polynomial of degree n or lower plus a little o.
In the case of an expansion at a real value of x, the formula pretty much covers it. For values of x approaching infinity, this thread pretty much covered it well (y= 1/x becomes the variable for the polynomial so technically the powers are still positive (for y))
We also studied what we call Generalized Limited Developments (I'm not sure what they're called properly in english), but they are defined as shown in the following image:GLD Definition
They are also defined at infinity (instead of near zero), the only difference is the multiplication of x to some power by f to get g, becomes division.
Now, my two main questions are:
- What are these 'Generalized Taylor Series' properly called in English? (A link to a course or wikipedia page is very much appreciated)
- In these expansion there are terms of x with positive powers and others with negative ones, so doesn't this mean that it's no longer a polynomial? and the original definition of a Taylor Expansion said it was a polynomial.
If you have any additional information about terminology, please tell me! Thanks.