Question

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

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

Hint:

When summing series involving factorials, use series expansion techniques and recognize known limits and convergence.

Answer 1

The Correct Option is A

To solve the limit, first break down the expression into manageable parts and recognize the factorial structure that allows for simplification using known summation formulas or the properties of factorials. After performing the simplification and taking the limit as $n \to \infty$, the value of the sum converges to $\frac{4}{3}$.