So I want an approximation $C$ for $\cos(1)$ by using the taylor expansion of the Cosinus-function. How many terms of the taylor expansion and how many decimal places do I need for $$ | C - \cos(1)| < 10^{-4}$$
I will rework the solution because there was a mistake and I want to give a solution which is more detailed.
2) You are correct, either one has to set $x=1$ on the left side, or the middle and right term are missing a factor $x^{2m+2}$, with $0<\nu<x$.
3) Adding smaller quantities repeatedly to $1$ incurs a floating point error of no more than the machine constant $\epsilon$ in each operation. One could argue that the computation of the term gives an relative error of at most $ϵ/2$ and the rounding after addition also at most $ϵ/2$ so that the combined effect is again an error of $ϵ$. As you add $m$ terms, the error of the floating point approximation is at most $mϵ$ larger than the error of the Taylor polynomial.