https://file.aiquickdraw.com/mathbot/file/1726712140467ymxsq9dg.jpg

The image you've uploaded represents a series with the general form:

\[
M = \sum_{n=1}^{\infty} \frac{\sqrt{n+1} - \sqrt{n}}{3 - \sqrt{n+1}}.
\]

To determine whether this series is **convergent** or **divergent**, we can apply some tests for convergence. The most common tests for series involve comparing the behavior of the terms \( a_n \) for large \( n \). Let's start with simplifying the general term and applying a test such as the **Limit Comparison Test** or **Ratio Test**.

Here’s a rough outline of how to approach this:

1. **Simplifying the terms**: 
   - The numerator \(\sqrt{n+1} - \sqrt{n}\) can be rationalized using a technique:
     \[
     \sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n+1} + \sqrt{n}}.
     \]
   - Therefore, the general term becomes approximately:
     \[
     a_n \approx \frac{1}{(3 - \sqrt{n+1})(\sqrt{n+1} + \sqrt{n})}.
     \]

2. **Asymptotic behavior**: For large \( n \), this will help to understand how the term behaves as \( n \to \infty \).

Would you like a detailed step-by-step analysis of this problem using a specific test, like the Limit Comparison Test or Ratio Test? Feel free to ask questions or request further explanation.

### Relative Questions:
1. Would you like a more detailed explanation of how to simplify the general term?
2. Should we apply the Ratio Test for convergence here?
3. Are you familiar with the Limit Comparison Test, and would you like to see it in action?
4. Would you like to explore alternative methods for checking series convergence?
5. Should we review some key convergence tests that apply to series like this?

### Tip:
When testing for convergence, simplifying the general term for large \( n \) often provides a clearer insight into the behavior of the series.

Is the series with general term \( \frac{\sqrt{n+1} - \sqrt{n}}{3 - \sqrt{n+1}} \) convergent or divergent?

Explore the convergence of the series with the general term \( \frac{\sqrt{n+1} - \sqrt{n}}{3 - \sqrt{n+1}} \). Learn about key concepts such as rationalizing square roots and using tests like the Limit Comparison and Ratio Test to determine the series' behavior as n approaches infinity. This topic is suitable for students at the undergraduate level, particularly those studying Calculus II.

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