Question

For any positive integer \( n \), the value of \( 1 + 1! + 2.2! + 3.3! + \cdots + n.n! \) will be

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

Hint:

When simplifying factorial series, break them into terms that allow easy summation.

Answer 1

The Correct Option is B

Step 1: Simplifying the given series.
The given series is: \[ 1 + 1! + 2.2! + 3.3! + \cdots + n.n! \] This is equivalent to: \[ \sum_{k=1}^{n} k \cdot k! \] which simplifies to: \[ \sum_{k=1}^{n} (k+1)! - 1 \] Step 2: Summing the series.
The sum of the first \( n \) terms of the series is: \[ (n+1)! - 1 \] Step 3: Conclusion.
The value of the series is \( n! + 1 \). The correct answer is (2) \( n! + 1 \).