In the Taylor series expansion included in the lecture slides, we consider f(x) = tk(x) + Rk+1(x). I am confused as to why in this question we have t4(x) and R6(x).
In this Taylor series, the odd terms are 0.
So the indexing I used is consistent with the respective degree of x,
and the associated derivative.
If you really like it better, instead of writing cos(x) = t4(x) + R6(x, ksi(x)),
you can write cos(x) = t4(x) + 0 + R6(x, ksi(x))
(You can also view t4+0 as t5, but the part of t5 we focus on is t4, as the other is 0)
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so R6 is not a single term?
R6 is a single term, it involves x and an unknown point
(often denoted by ksi or c).