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The series $\sum_{n=1}^\infty \frac{n}{n+1}$ is convergent because it satisfies the limit comparison test. Comparing it to the harmonic series $\sum_{n=1}^\infty \frac{1}{n}$, which is divergent, we have $\lim_{n\to\infty} \frac{\frac{n}{n+1}}{\frac{1}{n}} = \lim_{n\to\infty} \frac{n^2}{n+1} = \infty > 1$. Therefore, $\sum_{n=1}^\infty \frac{n}{n+1}$ is also divergent.