Question

The value of \[ \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^3 + 6k^2 + 11k + 5}{(k + 3)!} \right) \] is:

• A. \( \frac{4}{3} \)
• B. \( \frac{5}{3} \)
• C. 2
• D. \( \frac{7}{3} \)

Hint:

For sums involving factorials, focus on the dominant terms for large \( k \), and use asymptotic behavior to approximate the sum.

Answer 1

The Correct Option is A

Step 1: Given a sum of factorial terms, compute the limit as \( n \to \infty \). Simplify the expression by analyzing the asymptotic behavior of the sum.
Step 2: For large \( k \), \( (k + 3)! \) grows significantly faster than the polynomial terms in the numerator, causing the sum's terms to decrease rapidly as \( k \) increases. 
Step 3: The sum is primarily determined by its initial terms. Compute the infinite sum's value by summing these initial terms and evaluating the limit. The sum converges to \( \frac{4}{3} \). Therefore, the correct answer is (1).