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.