Approximate the value of the series to within an error of at most $10^{-4}$.
\[
\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}
\]
According to Equation (2):
\[
\left|S_{N}-S\right| \leq a_{N+1}
\]
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 \(\boxed{6}\).

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 \(\boxed{0.9720800630144033}\).