Find f(x) using concept of Definite-Integral-as a limit of sum

Question

$$ f(x) = \lim_{n\to\infty} \left[\frac{n^n\left(x^2+n^2\right)\left(x^2+\frac{n^2}{4}\right)\cdots\left(x^2+\frac{n^2}{n^2}\right)}{n!\left(x^3+n^3\right)\left(x^3+\frac{n^3}{8}\right)\cdots\left(x^3+\frac{n^3}{n^3}\right)}\right]^\frac{x}{n},\qquad x>0$$

My approach:

I took $\log$ on both sides.. $$ \log(f(x)) = x \lim_{n\to\infty}\left[ \sum_{r=1}^n \frac{1}{n} \log\left(x^2+\frac{1}{\left(\frac{r}{n}\right)^2}\right) - \sum_{r=1}^n \frac{1}{n} \log\left(x^3 + \frac{1}{(\frac{r}{n})^3}\right) + \frac{1}{n} \log\left(\frac{n^n}{n!}\right)\right]$$

The first two terms inside the square brackets become..

$$ \log(f(x)) = x \left[\int_0^1 \log\left(x^2+\frac{1}{t^2}\right) dt + \int_0^1 \log\left(x^3+\frac{1}{t^3}\right) dt + \lim_{n\to\infty} \frac{1}{n} \log\left(\frac{n^n}{n!}\right)\right]$$

I was unable to solve the integrals and the last term..Any Help ??

The last term is obviously $e$, because Stirling and stuff. As for the integrals, do them by parts. Also, your title is not very informative. — Ivan Neretin, Apr 20 '17 at 08:13
@ivan how last term is e?? i havent read about stirling and i did use by parts but it became complicated — Abhash Jha, Apr 20 '17 at 08:15
Your questions pretty much answer themselves. If you haven't read about something (like Stirling's approximation), then go and read about it. And if you don't like complicated things, well, then probably you shouldn't have come near this problem in the first place, because, well, just look at it... But then again, you already did the most challenging part. — Ivan Neretin, Apr 20 '17 at 08:19

score 1 · Accepted Answer · answered Apr 20 '17 at 08:40

Rather than first using riemann sum lets simplify some things. Take out $n^2,\frac {n^2}{4},...\frac {n^2}{n^2} $ from the brackets of numerator and $n^3,\frac {n^3}{8},....\frac {n^3}{n^3} $ from brackets of denominator. This converts the function as $$(\frac {n^n.n^{2n}.\frac {1}{1.4.9...n^2}(\frac {x^2}{n^2}+1)(\frac{2^2}{n^2}x^2+1)...(\frac {n^2}{n^2}x^2+1)}{n!.n^{3n}.\frac {1}{1^3.2^3...n^3}(\frac{x^3}{n^3}+1)(\frac {2^3}{n^3}x^3+1)... (\frac {n^3}{n^3}x^3+1)})^{x/n}$$ . Now remember that $\pi _1^n \frac {n^3}{n^2}=n!$ thus the function becomes $\frac {(\frac {x^2}{n^2}+1)(\frac {4x^2}{n^2}+1).... (\frac {n^2x^2}{n^2}+1)}{(\frac {x^3}{n^3}+1)(\frac {2^3x^3}{n^3}+1).. (\frac {n^3x^3}{n^3}+1)}^{\frac {x}{n}} $ now take logs and the $f (x)=\int _0 ^1 x\log (\frac {(xr)^2+1}{(xr)^3+1})dr $ now just put $rx=k $ note that x is constant and then integration can be done by separating logs and then by parts indivdually.

Find f(x) using concept of Definite-Integral-as a limit of sum

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