SOLUTION: 1(1!) + 2(2!) + 3(3!) + . . . + n(n!). Find the sum of this series if n = 15.

Question 572482: 1(1!) + 2(2!) + 3(3!) + . . . + n(n!).
Find the sum of this series if n = 15.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
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.
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