https://file.aiquickdraw.com/mathbot/file/17285178097433vo8m86u.jpg

The given series is:

\[
\sum_{n=1}^{\infty} \frac{n}{2n + 3}
\]

### Step 1: Analyze the Behavior of the Series

We need to check if this series converges or diverges. The form of the general term is:

\[
a_n = \frac{n}{2n + 3}
\]

For large \(n\), the behavior of this term is approximately:

\[
a_n \approx \frac{n}{2n} = \frac{1}{2}
\]

### Step 2: Apply the Test for Divergence

To check if the series diverges, we apply the **Test for Divergence**, which states that if the limit of the terms does not go to 0 as \(n \to \infty\), then the series diverges.

\[
\lim_{n \to \infty} \frac{n}{2n + 3} = \lim_{n \to \infty} \frac{1}{2 + \frac{3}{n}} = \frac{1}{2}
\]

Since the limit of the terms is \( \frac{1}{2} \), which is **not zero**, the series **diverges**.

### Conclusion:

- **Does the series converge?** No (it diverges).
- **Value if convergent:** Leave this blank since the series diverges.

Would you like any further clarification?

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Here are 5 related questions you could explore:
1. What is the Test for Divergence and when is it used?
2. How does the Limit Comparison Test work for series?
3. Can the Ratio Test be applied to determine the convergence of similar series?
4. What is the general behavior of a series with terms \( \frac{n}{an + b} \)?
5. How does the Integral Test help in determining the convergence of a series?

**Tip:** Always check the behavior of the general term of a series as \(n \to \infty\) before applying more complex convergence tests.

Consider the series ∑(n=1 to ∞) (n / (2n + 3)). Determine whether the series converges, and if it converges, determine its value.

Analyze whether the series ∑(n / (2n + 3)) converges using the Test for Divergence. The series diverges since the limit of its terms is not zero. A step-by-step solution is provided for undergraduate calculus students.

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