Taylor Expansions are polynomials based on the derivatives of a function that are used to approximate a function around a given point a.
Let f(x) be a function n-times differentiable in a neighborhood of (around) a point a. The n-th order Taylor Expansion(Taylor Polynomial) of the function f(x) around the point a is:
Maclaurin Expansions
Maclaurin Expansions(Maclaurin Polynomials) are the same as Taylor Expansions, or Taylor Polynomials, except that they're always around the point a=0.
The n-th order maclaurin expansion of a function is defined around the point a =0:
REMEMBER THAT THE (n) IN THE FUNCTION STANDS FOR THE ORDER OF THE DERIVATIVE, AND NOT A POWER!