I was trying to find the sum of the series $$\sum_{n=1}^\infty \frac {n} {(n+1)!}$$ I have found and proved by induction that a formula for the $k^{th}$ partial sum is $$S_k=\sum_{n=1}^k \frac {n} {(n+1)!}=\frac {(k+1)!-1} {(k+1)!}$$
I know that the sequence $S_1,S_2,S_3,\dots$ is an increasing sequence and I believe that the supremum of this sequence is $1$. So, I want to use the theorem that every monotone bounded sequence converges to its supremum to conclude that $$\sum_{n=1}^\infty \frac {n} {(n+1)!}=1$$
But I am having trouble showing that $1$ is the supremum of this sequence. Could someone please assist me in proving that the supremum of this sequence is $1$?