Given the Taylor series $$\sum_{i=0}^\infty (-1)^i\frac{(3n+1)}{2^n}(x-2)^i ,$$ if the $3$rd degree Taylor polynomial for $f$ centered at $x=2$ is used to approximate $f\left(\frac94\right)$, what is the alternating series error bound?"
Now I understand how to get the approximation $$T_3\left(\frac94\right)= 1-2\left(\frac94-2\right)+\frac74\left(\frac94-2\right)^2-\frac54\left(\frac94-2\right)^3 .$$
What I don't understand is how do I calculate the error bound without knowing the original function. I know the formula for the bound on $R_n(x)$, but it requires knowing $f(x)$, unless I am missing something.
"... what is the alternating series error bound?"
This is not asking for the Taylor series error. It is asking for the alternating series error. For a convergent alternating series, the sum of the series is bounded by any two successive partial sums. Thus, the $3^\text{rd}$ degree Taylor approximation at $9/4$ and the $4^\text{th}$ degree Taylor apprixmation at $9/4$ form an interval containing the value of the function at $9/4$. The error is bounded by the width of the interval.