_ n=1 ^m n n ! is equal to

Step-by-step solution

The given sum is S = _ n=1 ^ m n n! . Observe that the general term can be written as n n! = (n+1-1) n! = (n+1) n! - n! . Since (n+1) n! = (n+1)! , the term becomes (n+1)! - n! . The summation is a telescoping series: S = _ n=1 ^ m ((n+1)! - n!) S = (2! - 1!) + (3! - 2!) + (4! - 3!) + + ((m+1)! - m!) All intermediate terms cancel out, leaving only the last positive term and the first negative term: S = (m+1)! - 1! S = (m+1)! - 1 . Answer: (m+1) !-1