What is the sum

考慮通項 : n / (n + 1)!
= [(n + 1) - 1] / (n + 1)!
= (n + 1) / (n + 1)! - 1 / (n + 1)!
= 1 / n! - 1 / (n + 1)!
因此 :
1/2! + 2/3! + 3/4! +...+ 99/100!
=(1/1! - 1/2!) + (1/2! - 1/3!) + (1/3! - 1/4!) + ... + (1/99! - 1/100!)
= 1/1! - 1/100!
= 1 - 1/100!