Question 572482
1(1!) + 2(2!) + ... + 15(15!)


15(15!) can be rewritten as (16-1)15!, or 16! - 15!. Similarly, 14(14!) can be rewritten as 15! - 14!. This pattern continues and we can rewrite 1(1!) as 2! - 1!. Adding these up, we obtain


(2! - 1!) + (3! - 2!) + ... + (16! - 15!)


All the terms cancel, except the -1! and 16!. Therefore the series telescopes to 16! - 1!, or 16! - 1.