The value of the expression 3(1!) - 4(2!) + 5 | vedclass

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(C) Let the general term of the series be $T_n = (-1)^{n-1} (n+2)n!$. We can rewrite the terms as: $T_n = (-1)^{n-1} ((n+1) + 1)n! = (-1)^{n-1} ((n+1)

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$1$

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(C) Let the general term of the series be $T_n = (-1)^{n-1} (n+2)n!$. We can rewrite the terms as: $T_n = (-1)^{n-1} ((n+1) + 1)n! = (-1)^{n-1} ((n+1)! + n!)$. Expanding the series: $S = (2! + 1!) - (3! + 2!) + (4! + 3!) - (5! + 4!) + \dots - (2007! + 2006!) + 2007!$. Observing the telescoping nature of the sum: $S = 2! + 1! - 3! - 2! + 4! + 3! - 5! - 4! + \dots - 2007! - 2006! + 2007!$. All terms cancel out except for $1!$ and the last term. $S = 1! = 1$.

The value of the expression $3(1!) - 4(2!) + 5(3!) - 6(4!) + \dots - 2008(2006)! + (2007)!$ is:

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