Question

Let \( a_n = n + \frac{1}{n} \), \( n \in \mathbb{N} \). Then the sum of the series \( \sum_{n=1}^{\infty} (-1)^n n^{n+1 \frac{a_{n+1}}{n!} \) is}

• A. \( e - 1 \)
• B. \( e^{-1} \)
• C. \( 1 - e^{-1} \)
• D. \( 1 + e^{-1} \)

Hint:

When solving series with alternating terms and factorials, look for patterns related to the Taylor series expansions of exponential functions.

Answer 1

The Correct Option is D

Step 1: Understanding the sequence and series.
We are given the sequence \( a_n = n + \frac{1}{n} \), and we are asked to find the sum of the given series. First, note that the series involves alternating terms with \( (-1)^n \), and it appears to involve the exponential function \( e \).

Step 2: Rewriting the Series.
We rewrite the series and recognize that it has a form similar to the expansion of the exponential function \( e^x \). After simplifying, the sum of the series converges to \( 1 + e^{-1} \).

Step 3: Conclusion.
The correct answer is (D) \( 1 + e^{-1} \).