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zmudz
 one year ago
Find a closed form for
\(S_n = 1 \cdot 1! + 2 \cdot 2! + \ldots + n \cdot n!.\)
for integer \(n \geq 1.\) Your response should have a factorial.
zmudz
 one year ago
Find a closed form for \(S_n = 1 \cdot 1! + 2 \cdot 2! + \ldots + n \cdot n!.\) for integer \(n \geq 1.\) Your response should have a factorial.

This Question is Closed

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Notice that \[n\cdot n!=(n+11)\cdot n!=(n+1)\cdot n!n!=(n+1)!n!\] Now, \[\begin{align*}S_n&=1\cdot1!+2\cdot2!+\cdots+(n1)\cdot(n1)!+n\cdot n!\\[1ex] &=(2!1!)+(3!2!)+\cdots+(n!(n1)!)+((n+1)!n!)\\[1ex] &=\cdots \end{align*}\]
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