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: We are given a sum involving factorial terms, and we need to compute the limit as \( n \to \infty \). We can simplify the expression by examining the asymptotic behavior of the sum. 
Step 2: For large \( k \), the term \( (k + 3)! \) grows very quickly compared to the polynomial terms in the numerator. Thus, the terms of the sum decrease rapidly as \( k \) increases. 
Step 3: We recognize that the sum is dominated by the first few terms, and we compute the value of the infinite sum by summing the first few terms and taking the limit. The value of the sum as \( n \to \infty \) is \( \frac{4}{3} \). Thus, the correct answer is (1).