<s>[INST] Compute $\lim_{n \to +\infty} n^{-\frac12 \left(1+\frac{1}{n}\right)} \left(1^1 \cdot 2^2 \cdot 3^3 \cdots n^n \right)^{\frac{1}{n^2}}$ 
How to compute
  $$\displaystyle \lim_{n \to +\infty} n^{-\dfrac12 \left(1+\dfrac{1}{n}\right)} \left(1^1\cdot 2^2 \cdot 3^3 \cdots n^n \right)^{\dfrac{1}{n^2}}$$
  I'm interested in more ways of computing limit for this expression

My proof:
Let $u_n$be that sequence we've:
\begin{eqnarray*}
\ln u_n &=& -\frac{n+1}{2n}\ln n + \frac{1}{n^2}\sum_{k=1}^n k\ln k\\
&=& -\frac{n+1}{2n}\ln n + \frac{1}{n^2}\sum_{k=1}^n k\ln \frac{k}{n}+\frac{1}{n^2}\sum_{k=1}^n k\ln n\\
&=&  \frac{1}{n^2}\sum_{k=1}^n k\ln \frac{k}{n}\\
&=& \frac{1}{n}\sum_{k=1}^n \frac{k}{n}\ln \frac{k}{n}\\
&\to&\int_0^1 x\ln x\,dx = -1/4
\end{eqnarray*}
Therefore the limit is $e^{-\frac{1}{4}}$
 [/INST] Another approach, considering $$A_n= n^{-\dfrac12 \left(1+\dfrac{1}{n}\right)} \left(\prod_{i=1}^n i^i \right)^{\dfrac{1}{n^2}}=n^{-\dfrac12 \left(1+\dfrac{1}{n}\right)} H(n)^{\dfrac{1}{n^2}}$$ where appears the hyperfactorial function. Taking logarithms $$\log(A_n)={-\dfrac12 \left(1+\dfrac{1}{n}\right)}\log(n)+\dfrac{1}{n^2}\log(H(n))$$ Now, using Stirling like approximations $$\log(H(n))=n^2 \left(\frac{1}{2} \log(n)-\frac{1}{4}\right)+\frac{1}{2} n
   \log (n))+\left(\log (A)+\frac{1}{12} \log
   \left(n\right)\right)+O\left(\frac{1}{n^2}\right)$$ where appears Glaisher constant.
All of that, once recombined, leads to $$\log(A_n)=-\frac{1}{4}+\frac{\log (A)+\frac{1}{12} \log
   \left(n\right)}{n^2}+O\left(\frac{1}{n^{5/2}}\right)$$ which shows the limit and how it is approached.
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