Question

What is the sum of the first \( n \) terms of the series, whose \( k \)-term is \( k! \cdot k \)?

• A. \( (n+1)!^n - 1 \)
• B. \( (n+1)^n - 1 \)
• C. \( (n+1)! - 1 \)
• D. \( 3n - 2 \)

Hint:

Series Involving Factorials}
Try to rewrite factorial terms using telescoping pattern.
Recognize: \( k \cdot k! = (k+1)! - k! \).
Telescoping sum simplifies to first and last terms only.

Answer 1

The Correct Option is C

Given series: \[ S = 1! \cdot 1 + 2! \cdot 2 + 3! \cdot 3 + \cdots + n! \cdot n \] But \( k! \cdot k = k \cdot k! = (k+1)! - k! \) So, \[ S = \sum_{k=1}^{n} [(k+1)! - k!] = (n+1)! - 1! \Rightarrow S = (n+1)! - 1 \]