Answer
The value for $n=4$ and the remainder is less than $10^{-4}$.
Work Step by Step
We will use Remainder Alternating series such as: $\text{Error}=|R_n| \leq a_{n+1}$
Here, we have
$R_n \leq \dfrac{1}{(n+1)^6} \lt 10^{-4}$
and $ (n+1)^6 \gt 10^{4}$
or, $n+1 \gt \sqrt[6] {10000}$
or, $n \gt 3.64$
This implies that the value for $n=4$ and the remainder is less than $10^{-4}$.
