Question

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)!} = $

          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)!} = $

Consider the series ∑n=1^∞(n)/((n+1)!)
(a) Find the partial sums s1, s2, s3, and s4. Do you recognize the denominators?
s1 =
s2 =
s3 =
s4 =
(b) Use the pattern to guess a formula for sn
∘((n+1)! - 1)/((n+1)!)
∘((n+1)!)/((n+1)!)
∘((n+1)! + 1)/((n+1)!)
∘((n-1)! - 1)/((n-1)!)
∘((n+1)! - 1)/((n+1)!)
∘(n! - 1)/(n!)
∘(n - 1)/(n)
(c) Show that the given infinite series is convergent, and find its sum. (If the quantity diverges, enter DIVERGES.)
∑n=1^∞(n)/((n+1)!) =

Added by Erica A.

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