Question

The value of \( \Sigma_{r=1}^{n} \frac{1}{2^n nPr r!} \) is:

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

Hint:

Look for known series expansions when dealing with sums involving factorials and permutations.

Answer 1

The Correct Option is B

Step 1: Identify the terms in the series. We are given the sum: \[ S = \Sigma_{r=1}^{n} \frac{1}{2^n \cdot nP_r \cdot r!}. \] This is a series with factorial and permutation terms. It is a known series and can be related to expansions involving binomial coefficients. 
Step 2: Recognize the pattern. The sum resembles a known form involving the powers of 2, specifically a result from generating functions or exponential series. The general form of the sum leads to the conclusion: \[ S = 1 - 2^{-n}. \] Step 3: Conclude the result. Thus, the value of the series is: \[ \boxed{1 - 2^{-n}}. \]