Question

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

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

Hint:

For factorial series, try to write the term as a difference of two consecutive factorials. This often creates a telescoping series.

Answer 1

The Correct Option is C

Step 1: Write k * k! in telescoping form.
Observe \(k\cdot k!=(k+1)!-k!\). This is a telescoping series.

Step 2: Sum and cancel.
\(\sum_{k=1}^n k\cdot k!=\sum_{k=1}^n[(k+1)!-k!]=(n+1)!-1!=(n+1)!-1\). \[ \boxed{(n+1)!-1} \]