A series hits a wall

The gold curve is a polynomial copied from the curve at 0, term by term, up to the power xⁿ. For sin x, as n grows the stretch it hugs (where it stays within a pixel of the curve) reaches further out; from degree 27 it fills the whole frame (the readout shows ≥), and beyond the frame it goes on just the same: it reaches as far as you like. Switch to arctan x, just as smooth, and even with the slider all the way up the stretch only creeps towards 1; beyond degree 9, the error at 1.1 grows with every term. Now look in the complex plane: the colour at each point is the error, one contour for each power of ten, and the gold line is where the error is 1. arctan shoots to infinity at i and −i, two bad points, each one unit from 0; as n grows, the contours close in on the ice-blue circle through ±i, which is why on the real line the copy stops at 1. sin has no bad points at all: its gold line keeps moving out, and the view pulls back to follow it. The last choice works out, from the same coefficients, one polynomial divided by another; here n only decides which coefficients go in: the fraction uses them up to x to the power 4m − 1, and m goes up by 1 for every 4 added to n (from 1 to 10). In the complex plane, this fraction puts its bad points, the gold dots, on the imaginary axis beyond ±i, clear of the circle; once m is large it hugs the real line right across the picture. With n all the way up, its error at x = 2 is under one part in a hundred million, while the series with the same coefficients is out by more than ten billion. The wall belongs to copying at one point, term after term.