When I evaluate the following Chebyshev series of the first kind in two different ways, I get two very different results:
N[ChebyshevT[100, Cos[Pi/7]], 8] N[ChebyshevT[100, Cos[Pi/7]]] Out = 0.62348980 Out = 3.71097*10^18
Clearly the first result is the correct one. Could someone please explain why the difference occurs e.g. is it perhaps because of a peculiar way that the long Chebyshev polynomial is numerically evaluated? The problem does not arise for shorter length Chebyshev polynomials.