Consider the series $\sum_{n=1}^{\infty} \frac{n}{(n+1)!}$
(a) Find the partial sums $s_1, s_2, s_3,$ and $s_4$. Do you recognize the denominators?
$s_1 = $
$s_2 = $
$s_3 = $
$s_4 = $
(b) Use the pattern to guess a formula for $s_n$
$\circ \frac{(n+1)! - 1}{(n+1)!}$
$\circ \frac{(n+1)!}{(n+1)!}$
$\circ \frac{(n+1)! + 1}{(n+1)!}$
$\circ \frac{(n-1)! - 1}{(n-1)!}$
$\circ \frac{(n+1)! - 1}{(n+1)!}$
$\circ \frac{n! - 1}{n!}$
$\circ \frac{n - 1}{n}$
(c) Show that the given infinite series is convergent, and find its sum. (If the quantity diverges, enter DIVERGES.)
$\sum_{n=1}^{\infty} \frac{n}{(n+1)!} = $
