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    $\begingroup$ Heh, I'm not quite certain it makes sense to rationalize here, since the asymptotic value is unknown... the numerator and denominator could very well be anything :) For instance, it seems pretty close to $$\frac{5e}{\pi^5}\left(1+10^{-2}+10^{-4}\right)$$ :D $\endgroup$ Commented Sep 17, 2013 at 23:52
  • $\begingroup$ @rm-rf. You're right. But I only claimed what I showed here is "a way to get close" $\endgroup$ Commented Sep 18, 2013 at 0:10
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    $\begingroup$ Of course, your answer is helpful. Don't mind me here... I just needed a place to plug in my stupid "approximation" :P $\endgroup$ Commented Sep 18, 2013 at 0:12
  • $\begingroup$ @rm-rf. Actually, it's a good thing you did. That way some random reader don't take my post as "valid". Thanks for sharing the "approximation" $\endgroup$ Commented Sep 18, 2013 at 0:15