This is an alternating series problem. To approximate the sum with a given error bound, we can apply the Alternating Series Remainder Theorem, which states that the error $|R_n|$ in approximating the sum $S$ by the partial sum $S_n$ of an alternating series is bounded by the absolute value of the first omitted term:

$|R_n| = |S - S_n| \leq |a_{n+1}|$

In this case, we need the error to be less than $0.04$, so:

$|a_{n+1}| < 0.04$

The terms of the series are given by:

$a_n = \frac{(-1)^{n+1}}{\sqrt{2n + 1}}$

Thus:

$|a_n| = \frac{1}{\sqrt{2n + 1}}$

We now want the smallest $n$ such that:

$\frac{1}{\sqrt{2n + 1}} < 0.04$

Steps to solve for $n$:

1. Square both sides to eliminate the square root: $\frac{1}{2n + 1} < 0.04^2$

2. Simplify $0.04^2 = 0.0016$, giving: $2n + 1 > \frac{1}{0.0016} = 625$

3. Solve for $n$: $2n > 624 \Rightarrow n > 312$

Therefore, we need at least $n = 313$ terms to ensure the error is less than $0.04$.

Would you like a more detailed breakdown or have any other questions?

Follow-up Questions

1. What is the Alternating Series Remainder Theorem?
2. How can you tell if a series is convergent by looking at its terms?
3. Why does squaring both sides help solve this inequality?
4. How do we know that this series is alternating?
5. What other types of series have remainder bounds?

Tip

When dealing with error bounds in alternating series, always consider the first omitted term to quickly approximate how many terms are needed.