First slide

Sigma operation series

Question

The sum of first n terms of the series $1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + 4 \cdot 4! + \dots is$

Moderate

A

$(n + 1)! - 1$
B

$n! - 1$
C

$(n - 1)! - 1$
D

None of these

Solution

Let $S_{n} = 1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + 4 \cdot 4!$ $+ \dots + n \times n!$
$\Rightarrow S_{n} = (2 - 1) 1! + (3 - 1) 2! + (4 - 1) 3! + (5 - 1) 4! + \dots + [(n + 1) - 1] n! = (2 \cdot 1! - 1!) + (3 \cdot 2! - 2!) + (4 \cdot 3! - 3!) + (5 \cdot 4! - 4!) + \dots + [(n + 1) n! - n!] = (2! - 1!) + (3! - 2!) + (4! - 3!) + (5! - 4!) + \dots + [(n + 1)! - n!] = (n + 1)! - 1! = (n + 1)! - 1$

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