Answer
$$n=10$$
Work Step by Step
Recall the formula that $\text{Error}=|R_n| \leq a_{n+1}$
Here, we have
$R_n \leq \dfrac{1}{(n+1)^4} \lt 10^{-4}$
and $ (n+1)^4 \gt 10^{4}$
or, $(n+1)^4 \gt 10^4$
or, $n \gt 10-1$
or, $n \gt 9$
or, $n=10$
