To solve this problem using the alternating series error estimate, we first need to recall the formula for the error bound in an alternating series.

For an alternating series of the form:

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

where $a_n$ is a decreasing sequence of positive terms that approach 0, the error after taking the partial sum $S_m$ (the sum of the first $m$ terms) is less than or equal to the first omitted term:

$|S - S_m| \leq |a_{m+1}|$

We are given the series:

$S = \sum_{n=1}^{\infty} \frac{(-1)^n}{5n^{3/2}}$

In this case, $a_n = \frac{1}{5n^{3/2}}$, and we want to find the smallest $m$ such that the error is less than 0.01, i.e.,

$|a_{m+1}| = \frac{1}{5(m+1)^{3/2}} < 0.01$

Step 1: Set up the inequality

We need:

$\frac{1}{5(m+1)^{3/2}} < 0.01$

Step 2: Solve for $m$

Multiply both sides by 5 to eliminate the denominator:

$\frac{1}{(m+1)^{3/2}} < 0.05$

Now, invert both sides:

$(m+1)^{3/2} > \frac{1}{0.05} = 20$

Take the 2/3 power of both sides to solve for $m+1$:

$m+1 > 20^{2/3}$

Using a calculator:

$20^{2/3} \approx 8.385$

So:

$m+1 > 8.385$

Thus:

$m > 7.385$

Therefore, the smallest integer $m$ is $m = 8$.

Conclusion:

The smallest $m$ such that the error between the partial sum and the actual sum is less than 0.01 is $m = 8$.

Would you like more details on how we arrived at this result? Or do you have any questions about this process?

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Related Questions:

1. How do alternating series behave in terms of convergence?
2. Can this method be applied to non-alternating series?
3. How do you calculate the sum of an alternating series directly?
4. What happens if the series is not decreasing or the terms do not approach zero?
5. How can we use this error estimate to approximate the actual sum more accurately?

Tip: Always check the conditions for using the alternating series error estimate: the terms must be positive, decreasing, and tend to zero as $n \to \infty$.