To approximate the value of the series to within an error of at most $10^{-5}$, we can use the alternating series error estimation. The given series is:

$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^7}$

For an alternating series with terms $a_n = \frac{1}{n^7}$, the error bound of a partial sum $S_N$ approximating the series sum $S$ is given by the magnitude of the first omitted term:

$|S_N - S| \leq a_{N+1}$

We want this error to be within $10^{-5}$:

$a_{N+1} = \frac{1}{(N+1)^7} \leq 10^{-5}$

To find the smallest value of $N$ that satisfies this inequality, we can solve:

$(N+1)^7 \geq 10^5$

Taking the seventh root on both sides:

$N+1 \geq 10^{5/7}$

Calculating $10^{5/7}$ gives approximately 10.68. Therefore, we need $N+1 \geq 11$, so $N \geq 10$.

Thus, the smallest value of $N$ that approximates $S$ to within an error of at most $10^{-5}$ is $N = 10$.

Would you like further details, or do you have any questions?

Here are some related questions that can deepen your understanding:

1. How does the alternating series test determine convergence of the series?
2. Why is the error in an alternating series estimate given by the first omitted term?
3. What would change if the series was not alternating?
4. How would the approximation change if we required a smaller error, say $10^{-6}$?
5. What happens to the error bound as $N$ increases?

Tip: For alternating series, the error can often be quickly estimated by the next term, which simplifies error analysis.