You may want to look up Taylor series.

Recall that the nth order taylor polynomial of f(x) centered at a=0 is

Tn(x) = f(a) + f(a)(x-a) + (1/2!)f(a)(x-a) + (1/3!)f(a)^3 + ... + (1/n!)f(n)(a)(x-a) ^n Tnx)=f(a)+f(a)(xa)+12!f(a)(xa)+13!f(a)(xa)3++1n!f(n)(

These are the partial sums of the taylor series of f(x) centered at a. If a=0, this is often called the MacLaurin series.

In this problem, well fix a=0. Starting with a function f(x), we want to produce an animated in python so that the k-th frame displays the graph f(x) together with the graph Tk(x). We want to visualize here the taylor series converges to the function, i.e, for which xs does Tn(x) > f(x) as n > infinity.

Try these functions: e^x, sin(x), 1/(1-x), 1/(1+x^2), arctan(x), ln(1+x). State your observations about convergence of the taylor series in these cases.