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To test the convergence of the series:

\[
\sum_{n=1}^{\infty} (-1)^n \frac{\sqrt{n}}{1 + 2\sqrt{n}}
\]

we can analyze it as an **alternating series** (due to the \( (-1)^n \) factor) and check for convergence by using the **Alternating Series Test (Leibniz's Test)** and/or other tests if necessary.

### Step 1: Applying the Alternating Series Test

For an alternating series \( \sum (-1)^n a_n \), the series converges if:
1. \( a_n \) is positive.
2. \( a_n \) is decreasing.
3. \( \lim_{n \to \infty} a_n = 0 \).

In this case, let:
\[
a_n = \frac{\sqrt{n}}{1 + 2\sqrt{n}}
\]

#### 1. Positivity
Since \( \sqrt{n} \) and \( 1 + 2\sqrt{n} \) are both positive for \( n \geq 1 \), \( a_n > 0 \) for all \( n \geq 1 \).

#### 2. Monotonic Decrease
To check if \( a_n \) is decreasing, we could examine \( a_{n+1} < a_n \) or analyze the derivative if it’s a continuous function. Here, as \( n \) increases, both \( \sqrt{n} \) and \( 1 + 2\sqrt{n} \) increase, but the denominator grows faster than the numerator, suggesting that \( a_n \) decreases.

#### 3. Limit of \( a_n \) as \( n \to \infty \)
We compute the limit:
\[
\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{\sqrt{n}}{1 + 2\sqrt{n}}
\]
Dividing the numerator and denominator by \( \sqrt{n} \):
\[
= \lim_{n \to \infty} \frac{1}{\frac{1}{\sqrt{n}} + 2} = \frac{1}{0 + 2} = \frac{1}{2}
\]

Since \( \lim_{n \to \infty} a_n \neq 0 \), the series **fails the Alternating Series Test**.

### Step 2: Applying the Divergence Test

The Divergence Test states that if \( \lim_{n \to \infty} a_n \neq 0 \), then the series diverges. Since we found:
\[
\lim_{n \to \infty} a_n = \frac{1}{2} \neq 0
\]
the series **diverges**.

### Conclusion
The series \(\sum_{n=1}^{\infty} (-1)^n \frac{\sqrt{n}}{1 + 2\sqrt{n}}\) diverges.

---

Would you like further details on any of the steps? 

Here are some related questions for further exploration:

1. How does the Alternating Series Test work in general?
2. When should the Divergence Test be applied before other tests?
3. What if the limit of \(a_n\) goes to zero but the terms are not decreasing—does the series converge?
4. How does the growth rate of \( \sqrt{n} \) compared to \( 1 + 2\sqrt{n} \) affect the convergence?
5. Can you use other tests, like the Ratio Test, on this series?

**Tip:** Always check the Divergence Test as a first step; if \(\lim_{n \to \infty} a_n \neq 0\), the series cannot converge.

Test for convergence of the series \( \sum_{n=1}^{\infty} (-1)^n \frac{\sqrt{n}}{1 + 2\sqrt{n}} \).

Learn how to test the convergence of the series \( \sum_{n=1}^{\infty} (-1)^n \frac{\sqrt{n}}{1 + 2\sqrt{n}} \) using the Alternating Series Test and Divergence Test. Detailed step-by-step solution included, suitable for college-level students studying engineering mathematics.

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