We are asked to evaluate the infinite sum $\sum_{n=1}^{\infty} \frac{n}{n^2 + 2n + 3}$.

1. Problem Description

We are asked to evaluate the infinite sum $\sum_{n=1}^{\infty} \frac{n}{n^2 + 2n + 3}$.

2. Solution Steps

Let $a_n = \frac{n}{n^2 + 2n + 3}$.

We can compare $a_n$ with $b_n = \frac{n}{n^2} = \frac{1}{n}$. We know that $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges (harmonic series).

Let's consider the limit comparison test.

$$\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{n}{n^2 + 2n + 3}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n^2}{n^2 + 2n + 3} = \lim_{n \to \infty} \frac{1}{1 + \frac{2}{n} + \frac{3}{n^2}} = \frac{1}{1 + 0 + 0} = 1$$

Since the limit is a positive finite number (1), the series $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ either both converge or both diverge.

Since $\sum_{n=1}^{\infty} b_n = \sum_{n=1}^{\infty} \frac{1}{n}$ diverges (harmonic series), the series $\sum_{n=1}^{\infty} \frac{n}{n^2 + 2n + 3}$ also diverges.

3. Final Answer

The series diverges.

Diverges