<s>[INST] Limit of the $n$th root of the product How one can evaluate $$\lim_{n\to\infty} \frac{\left( \prod_{k=1}^n (n^2+k^2) \right)^{1/n}}{n^2}?$$ I was unable to found any trick to do the computation.
 [/INST] $$\frac{\left( \prod_{k=1}^n (n^2+k^2) \right)^{1/n}}{n^2}=\frac{\left( \prod_{k=1}^n (n^2(1+\frac{k^2}{n^2}) \right)^{1/n}}{n^2}= \frac{n^2 \left( \prod_{k=1}^n (1+\frac{k^2}{n^2} \right)^{1/n}}{n^2}$$
$$\log\frac{n^2 \left( \prod_{k=1}^n (1+\frac{k^2}{n^2} \right)^{1/n}}{n^2}= \sum_{k=1}^n \frac{1}{n}\log (1+\frac{k^2}{n^2})$$
$$\int_0^1 \log(1+x^2)dx =\lim \sum_{k=1}^n \frac{1}{n}\log (1+\frac{k^2}{n^2})$$
The last integral can be evaluated by parts: it equals $-2 + \frac{\pi}{2}+\log(2)\approx 0.2639435$.
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