Approximate the value of the series to within an error of at most ${10}^{-4}$.
$$\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n^5}$$
According to Equation (2):
$$|S_N-S|\le a_{N+1}$$
what is the smallest value of $N$ that approximates $S$ to within an error of at most ${10}^{-4}$ ?
$$N=$$
$$S\approx$$

Step 1 ：The given series is an alternating series. The error in approximating the sum of an alternating series by the sum of the first N terms is less than or equal to the absolute value of the (N+1)th term. So, we need to find the smallest N such that the (N+1)th term is less than or equal to ${10}^{-4}$.

Step 2 ：By calculating, we find that the smallest value of N that approximates S to within an error of at most ${10}^{-4}$ is $\boxed{{\displaystyle 6}}$.

Step 3 ：Now, let's calculate the approximate value of the series sum up to the 6th term.

Step 4 ：By calculating, we find that the approximate value of the series sum up to the 6th term is $\boxed{{\displaystyle 0.9720800630144033}}$.