9. [7] Evaluate the limit $\lim _{n \rightarrow \infty} n^{-\frac{1}{2}\left(1+\frac{1}{n}\right)}\left(1^{1} \cdot 2^{2} \cdots n^{n}\right)^{\frac{1}{n^{2}}}$.
Answer: $e^{-1 / 4}$ Taking the logarithm of the expression inside the limit, we find that it is
$$
-\frac{1}{2}\left(1+\frac{1}{n}\right) \ln n+\frac{1}{n^{2}} \sum_{k=1}^{n} k \ln k=\frac{1}{n} \sum_{k=1}^{n} \frac{k}{n} \ln \left(\frac{k}{n}\right) \text {. }
$$

We can recognize this as the as Riemann sum expansion for the integral $\int_{0}^{1} x \ln x d x$, and thus the limit of the above sum as $n \rightarrow \infty$ equals to the value of this integral. Evaluating this integral using integration by parts, we find that
$$
\int_{0}^{1} x \ln x d x=\left.\frac{1}{2} x^{2} \ln x\right|_{0} ^{1}-\int_{0}^{1} \frac{x}{2} d x=-\frac{1}{4}
$$

Therefore, the original limit is $e^{-1 / 4}$.