Find the Sum

$\sum_{n=1}^{\infty} \frac{n}{n+1}$ this series diverges by the limit comparison test.

Intuitively you can think of $\displaystyle \sum_{n \ = \ 1}^{\infty} \frac{n}{n + 1}$ as adding numbers whose limiting value as $n \to \infty$ is $1$ . At the infinite level this will get you infinitesimally close to adding $1$ s to infinity, and it is easy to see that $\displaystyle \sum_{n \ = \ 1}^{\infty} 1$ diverges to infinity. Rigorously speaking, the sequence $\displaystyle {a_n} = \frac 12, \frac 23, \frac 34, \frac 45, \cdots$ is monotonically increasing, so the sum $\displaystyle \sum_{n \ = \ 1}^{\infty} a_n$ is unbounded to infinity.