Answer
If $~~n \geq 347,346~~$ then $~~R_n-A \lt 0.0001$
Work Step by Step
We need to find a value of $n$ such that:
$R_n-A \lt \frac{b-a}{n}~[f(b)-f(a)] \lt 0.0001$
We can find a value of $n$:
$\frac{b-a}{n}~[f(b)-f(a)] \lt 0.0001$
$\frac{3-1}{n}~[f(3)-f(1)] \lt 0.0001$
$\frac{2}{n}~(e^3-e) \lt 0.0001$
$\frac{1}{n} \lt \frac{0.0001}{2(e^3-e)}$
$n \gt \frac{2(e^3-e)}{0.0001}$
$n \gt 347,345.1$
Since $n$ must be a whole number, $n \geq 347,346$
If $~~n \geq 347,346~~$ then $~~R_n-A \lt 0.0001$
