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.

1. Analyze (\sum_{n=1}^\infty \