Question

The value of $1+1.1!+2.2!+3.3!+.....+n.n!$ is

• A. $(n+1)!+1$
• B. $(n-1)!+1$
• C. $(n+1)!-1$
• D. $(n+1)!$

Answer 1

The correct option is D $(n+1)!$

$1+1.1!+2.2!+3.3!+.....+n.n!=1+\sum _{r=1}^{n}r.r!=1+\sum _{r=1}^n(r+1-1)r!=1+\sum _{r=1}^n(r+1)r!-r!=1+\sum _{r=1}^n(r+1)!-r!=1+(2!-1!)+(3!-2!)+(4!-3!)+.....+((n+1)!-n!)=(n+1)!$