Evaluate $\lim_{n\to\infty}{\sum_{0 \leq k \leq n}} \binom{n}{k}\frac{1}{n^{k}(k+3)}$

Question

Evaluate $$\lim_{n\to\infty}{\sum_{0 \leq k \leq n}} \binom{n}{k}\frac{1}{n^{k}(k+3)}.$$

My work: I tried using sum of binomial coefficients. I could not find a general term to find the limit.

Answer 1

Hint. Notice that, by the binomial theorem, $$\begin{align} \sum_{0 \leq k \leq n} \binom{n}{k}\frac{1}{n^{k}(k+3)}&= \sum_{0 \leq k \leq n} \binom{n}{k}\frac{1}{n^k}\int_{0}^1x^{k+2}dx \\&= \int_{0}^1x^2\sum_{0 \leq k \leq n} \binom{n}{k}\left(\frac{x}{n}\right)^kdx\\ &=\int_{0}^1x^2\left(1+\frac{x}{n}\right)^n\,dx. \end{align}$$ Then use Show that $f_n=(1+\frac{x}{n})^n$ converge uniformly on all compact of $\mathbb R$.?