Approximating arctan x by power series

Execute this piece of code by clicking anywhere in the code cell and pressing Enter.

Clear[x];
approximations = {
x, 
x - x^3 / 3, 
x - x^3 / 3 + x^5 / 5, 
x - x^3 / 3 + x^5 / 5 - x^7 / 7, 
x - x^3 / 3 + x^5 / 5 - x^7 / 7 + x^9 / 9 , 
x - x^3 / 3 + x^5 / 5 - x^7 / 7 + x^9 / 9 - x^11 / 11, 
x - x^3 / 3 + x^5 / 5 - x^7 / 7 + x^9 / 9 - x^11 / 11 + x^13 / 13, 
x - x^3 / 3 + x^5 / 5 - x^7 / 7 + x^9 / 9 - x^11 / 11 + x^13 / 13 - x^15 / 15
};

Then execute this next piece of code to make a visualizer. As you move the slider with your mouse, you can see how the power series for arctan x approximates it better and better --- but only on the interval (-1, 1).

Manipulate[
Plot[{ArcTan[x], approximations[[i]]}, {x, -2, 2}, PlotRange->{-4, 4}], 
{i, 1, Length[approximations], 1}]

Here are all of those approximations at once, superimposed on arctan x.

Plot[Append[approximations, ArcTan[x]], {x, -2, 2}, PlotRange->{-4, 4}]