To determine the convergence of the series

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

using the Ratio Test, we proceed as follows:

1. Identify $a_n$: The general term of the series is $a_n = \frac{(-1)^n \sqrt{n}}{1 + 2\sqrt{n}}.$

2. Apply the Ratio Test: The Ratio Test states that for a series $\sum a_n$, if $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$ then:

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

3. Compute $\left| \frac{a_{n+1}}{a_n} \right|$:

   = \frac{\sqrt{n+1}}{\sqrt{n}} \cdot \frac{1 + 2\sqrt{n}}{1 + 2\sqrt{n+1}}.$$

4. Simplify the expression: $\frac{\sqrt{n+1}}{\sqrt{n}} = \sqrt{\frac{n+1}{n}} = \sqrt{1 + \frac{1}{n}}.$ As $n \to \infty$, $\sqrt{1 + \frac{1}{n}} \to 1$.

   For the second part, $\frac{1 + 2\sqrt{n}}{1 + 2\sqrt{n+1}} \to 1 \quad \text{as } n \to \infty.$

5. Calculate the limit $L$: Combining these, we get $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1.$

Since $L = 1$, the Ratio Test is inconclusive for this series.

Would you like further details on other convergence tests that might work here?

Related Questions:

1. What is the Alternating Series Test and can it be applied here?
2. How does the Root Test compare to the Ratio Test for convergence?
3. Can absolute convergence and conditional convergence affect the result?
4. What other series tests are effective for terms with square roots?
5. How can we determine convergence without the Ratio Test?

Tip:

For alternating series, check if the terms decrease in absolute value and approach zero for potential conditional convergence.