To determine the convergence of the series using the ratio test, let's analyze the given series:

$\sum_{n=1}^{\infty} \frac{(-1)^n \sqrt{n}}{1 + \sqrt{n}}$

The ratio test states that for a series $\sum a_n$, if we compute the limit

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|,$

then:

1. If $L < 1$, the series converges absolutely.
2. If $L > 1$, the series diverges.
3. If $L = 1$, the test is inconclusive.

Step-by-Step Solution

1. Identify $a_n$: Here, $a_n = \frac{(-1)^n \sqrt{n}}{1 + \sqrt{n}}$.

2. Calculate $a_{n+1}$: $a_{n+1} = \frac{(-1)^{n+1} \sqrt{n+1}}{1 + \sqrt{n+1}}.$

3. Form the Ratio $\frac{a_{n+1}}{a_n}$: $\frac{a_{n+1}}{a_n} = \frac{\frac{(-1)^{n+1} \sqrt{n+1}}{1 + \sqrt{n+1}}}{\frac{(-1)^n \sqrt{n}}{1 + \sqrt{n}}}.$

   Simplifying, we get: $\frac{a_{n+1}}{a_n} = \frac{(-1)^{n+1} \sqrt{n+1} \cdot (1 + \sqrt{n})}{(-1)^n \sqrt{n} \cdot (1 + \sqrt{n+1})}.$

   Since $(-1)^{n+1} / (-1)^n = -1$, we obtain: $\frac{a_{n+1}}{a_n} = -\frac{\sqrt{n+1} (1 + \sqrt{n})}{\sqrt{n} (1 + \sqrt{n+1})}.$

4. Take the Limit as $n \to \infty$: $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{\sqrt{n+1} (1 + \sqrt{n})}{\sqrt{n} (1 + \sqrt{n+1})}.$

   To simplify this expression, divide the numerator and the denominator by $\sqrt{n}$: $L = \lim_{n \to \infty} \frac{\sqrt{\frac{n+1}{n}} (1 + \sqrt{\frac{n}{n}})}{1 + \sqrt{\frac{n+1}{n}}}.$

   As $n \to \infty$, $\sqrt{\frac{n+1}{n}} \to 1$ and $\sqrt{\frac{n}{n}} = 1$, so we have: $L = \frac{1 \cdot (1 + 1)}{1 + 1} = \frac{2}{2} = 1.$

Since $L = 1$, the ratio test is inconclusive for this series.

Would you like to explore alternative convergence tests, such as the alternating series test or comparison test, to further analyze this series?

Follow-Up Questions

1. What alternative convergence tests can we apply to this series?
2. Why is the ratio test inconclusive when $L = 1$?
3. How does the alternating factor $(-1)^n$ affect the convergence of this series?
4. Can the comparison test be used to analyze the behavior of this series?
5. Is it possible to simplify the series expression further to make other tests easier?

Tip

For series with alternating terms, consider the alternating series test, especially if the ratio test is inconclusive.