https://file.aiquickdraw.com/mathbot/file/1730498019564lm3lw70b.jpg

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.

Use the ratio test to determine the convergence of the series \( \sum_{n=1}^{\infty} \frac{(-1)^n \sqrt{n}}{1 + \sqrt{n}} \).

Learn how to apply the ratio test to determine the convergence of an alternating series with square roots. This step-by-step solution explains the ratio test formula and evaluates the limit to conclude whether the series \( \sum_{n=1}^{\infty} \frac{(-1)^n \sqrt{n}}{1 + \sqrt{n}} \) converges.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation