To solve this problem, we need to find the number of terms $N$ required to estimate the value of the series

$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^{3/2}}$

to within an error of $10^{-6}$.

Since this is an alternating series, we can use the Alternating Series Estimation Theorem. For an alternating series of the form

$\sum_{n=1}^{\infty} (-1)^{n-1} a_n$

where $a_n > 0$, the error $R_N$ after $N$ terms is bounded by the next term:

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

where $S$ is the actual sum of the series, and $S_N$ is the sum of the first $N$ terms.

In this case, $a_n = \frac{1}{n^{3/2}}$. We want the error to be less than $10^{-6}$, so:

$\frac{1}{(N+1)^{3/2}} < 10^{-6}$

Now, we can solve for $N$ by rearranging the inequality:

$(N+1)^{3/2} > 10^6$

Taking the cube root of both sides:

$N+1 > (10^6)^{2/3}$

Calculating $(10^6)^{2/3}$:

$10^{6 \times \frac{2}{3}} = 10^4 = 10000$

So:

$N+1 > 10000$

Thus, $N > 9999$.

Therefore, we need at least $N = 9999$ terms to estimate the value of the series to within $10^{-6}$.

Would you like more details on this solution or have any questions?

Here are five related questions:

1. How does the Alternating Series Estimation Theorem apply to other series?
2. Can this estimation method be applied to non-alternating series?
3. What happens if the convergence rate is slower than $\frac{1}{n^{3/2}}$?
4. How would this approach change if we needed an error bound of $10^{-8}$ instead?
5. Could we apply numerical methods to find the sum instead?

Tip: When working with series, check if it’s alternating, as this can provide useful bounds on the error through the Alternating Series Estimation Theorem.