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The exercise provided asks to study the convergence or divergence of three infinite series using the theorem of comparison to an integral. Let's analyze each one in detail:

### 1. \(\sum_{n=1}^\infty \frac{n}{n^2 + 1}\)
- **Rewrite the term:** \(\frac{n}{n^2 + 1} \approx \frac{n}{n^2} = \frac{1}{n}\) for large \(n\).
- **Comparison to \(\int \):** Compare it with the harmonic series \(\sum \frac{1}{n}\), which diverges.
- Perform integral testing on \(\int \frac{n}{n^2+1} dn\) to confirm if divergence matches.
  
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### 1. Analyze \(\sum_{n=1}^\infty \

Study the nature of the following series using the theorem of comparison to an integral: (1) sum(n/(n^2 + 1)) from n=1 to infinity, (2) sum(1/(n ln(n) ln(ln(n)))) from n=2 to infinity, (3) sum(1/(n^2 + 1)) from n=1 to infinity.

Explore the convergence or divergence of three infinite series using the integral comparison test. This problem evaluates advanced calculus concepts such as infinite series, harmonic series, and logarithmic functions. Suitable for undergraduate mathematics students.

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