If $\lim_{n \to \infty} \sum_{k=1}^n \frac{{^n}C_k}{n^k(k+3)} = p$ then $p$ is ____.
Step-by-Step Solution
Key Concept: Recognize binomial expansion in the sum and convert it to an integral using the limit definition of $e^x$.
The limit $\lim_{n \to \infty} \sum_{k=0}^{n} \frac{C_k}{n^k(k+3)}$ is evaluated by recognizing that $\sum_{k=0}^{n} C_k \left(\frac{x}{n}\right)^k = \left(1 + \frac{x}{n}\right)^n \to e^x$ as $n \to \infty$. The sum is converted to an integral: $\int_0^1 x^2 \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n dx = \int_0^1 x^2 e^x dx$. Using integration by parts twice yields $\left[x^2 e^x\right]_0^1 - \int_0^1 2xe^x dx = e - 2$.
Correct Answer: 0.72