The value of 1 2 ! + 2 3 ! + + 99 100 ! is equal to

Step-by-step solution

The general term of the series is T_n = n (n+1)! for n = 1, 2, , 99 . We can rewrite the general term as T_n = n+1-1 (n+1)! = n+1 (n+1)! - 1 (n+1)! = 1 n! - 1 (n+1)! . The sum S is given by _ n=1 ⁹⁹ ( 1 n! - 1 (n+1)! ) . This is a telescoping series: S = ( 1 1! - 1 2! ) + ( 1 2! - 1 3! ) + + ( 1 99! - 1 100! ) . All intermediate terms cancel out, leaving S = 1 1! - 1 100! = 1 - 1 100! . S = 100! - 1 100! . Answer: 100 !-1 100 !