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The image appears to show a webpage with a mathematical question. To proceed, I'll analyze the content and help you with the problem presented. Give me a moment to extract the relevant details.It seems that the question on the webpage asks:

**Question**:  
"Determine whether the series converges or diverges. If it converges, find the sum."  
\[
\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}
\]

Let me guide you through this:

1. **Step 1**: We have the infinite series \( \sum_{n=1}^{\infty} \frac{1}{n^2 + 1} \). To check whether it converges or diverges, we can apply a convergence test.

2. **Step 2**: We can use the **comparison test** or **limit comparison test**. Compare it to the simpler series \( \frac{1}{n^2} \), which is a convergent **p-series** with \( p = 2 \) (since \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \)).

3. **Step 3**: Since \( \frac{1}{n^2 + 1} \) is less than \( \frac{1}{n^2} \) for large \( n \), and \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges, by the **comparison test**, the given series also converges.

However, finding the sum analytically is not straightforward, as this is not a standard series for which a simple closed form exists. The series converges, but the sum is not elementary.

Would you like to go deeper into any of these steps?

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Here are 5 related questions:

1. What is the difference between absolute and conditional convergence?
2. How does the p-series test work, and when can it be applied?
3. Can the ratio test be used to determine convergence of this series?
4. What are some common comparison series used for convergence testing?
5. How does the integral test help in checking series convergence?

**Tip**: When testing series for convergence, always check if comparison with a known convergent series or a simpler p-series test can simplify the problem.

Determine whether the series converges or diverges. If it converges, find the sum.

Explore the convergence or divergence of the infinite series \( \sum_{n=1}^{\infty} \frac{1}{n^2 + 1} \) using the comparison test. This detailed step-by-step solution covers essential concepts like the p-series test and comparison test. Ideal for college-level mathematics students.

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