Taylor formula and McLaughlin formula

The Taylor formula (Taylor series) expresses a function as the sum of its successive derivatives at a certain point. If the function has n derivative at the point x=x0 _{, then f(x) can be expanded as:}

Its purpose is to approximate the function with a polynomial function. If x ₀ =0, it is the Maclaurin formula:

The larger the value of n, the closer it is to f(x). The following figure shows the 5th-order, 20th-order and 50th-order expansion of f(x)=sin ² x+cos ³ x at x ₀ =0. It can be seen that the more the value of n is. The larger it is, the closer it is to f(x).