Series with factorial.

Hint $$\frac{n}{(n+1)!}= \frac{n+1 -1}{(n+1)!} =\frac{1}{n!} - \frac{1}{(n+1)!}$$

This is a classic example of a telescoping series! It'd first be nice to shift the limits of summation to get the factorial of a more comfortable function, and then we can split up the fraction to show how it telescopes:

$$\begin{align} \sum_{n=1}^{99} \frac{n}{(n+1)!} &= \sum_{n=2}^{100} \frac{n-1}{n!} \\ &= \sum_{n=2}^{100} \frac{1}{(n-1)!} - \frac{1}{n!} \\ &= \frac{1}{(2-1)!} - \frac{1}{100!} \\ &= 1- \frac{1}{100!} \end{align}$$

As an aside, note that the result of this series converges towards $1$ as $n$ approaches infinity.