<s>[INST] A limit with infinite product $\lim\limits_{n\to\infty}\frac1{n^4}\prod_{i=1}^{2n}\left(n^2+i^2\right)^{\frac1n}$ How to calculate : 
$$\lim_{n\to\infty}\frac{1}{n^4}\prod_{i=1}^{2n}\left(n^2+i^2\right)^{\frac{1}{n}}$$
 [/INST] Pull the $n$ out of the product to get
$$L = \lim_{n \rightarrow \infty} \prod_{i=1}^{2 n} \left ( 1+\frac{i^2}{n^2} \right )^\frac{1}{n} $$
Rewrite as 
$$\log{L} =  \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{2 n} \left ( 1+\frac{i^2}{n^2} \right )$$
This is a Riemann sum whose limit is
$$\begin{align}\log{L} &= \int_0^2 dx \: \log{(1+x^2)}\\ &= [x \log{(1+x^2)}]_0^2 - 2 \int_0^2 dx \: \frac{x^2}{1+x^2} \\ &= 2 \log{5} + 2 \arctan{2} - 4 \end{align}$$
Then the limit $L$ is
$$L = 25 e^{-2 (2-\arctan{2})} $$
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