# Maths Dopers - Question for you! **URL:** \ **Category:** Factual Questions\ **Created:** [July 23, 2004, 9:47pm UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585 "2004-07-23T21:47:58Z")\ **Posts on this page:** 10\ **Page:** 1
**Author:** ![InvidiousCourgette](https://avatars.discourse-cdn.com/v4/letter/i/e47c2d/32.png) [@InvidiousCourgette](https://boards.straightdope.com/u/InvidiousCourgette)\ **Post date:** [July 23, 2004, 9:47pm UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585/1 "2004-07-23T21:47:58Z")
If you consider the formula: (x+1/2)^(x+1/2) = k. x!.e^x As x gets large k quickly moves towards a value of around 0.658 something. Is it possible to say what value k takes as x tends to infinity? Is it a special number, like pi/4 or what have you? ---
**Author:** ![Strainger](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/strainger/32/244_2.png) [@Strainger](https://boards.straightdope.com/u/Strainger)\ **Post date:** [July 23, 2004, 10:06pm UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585/2 "2004-07-23T22:06:30Z")
Are the dots supposed to indicate multiplication? ---
**Author:** ![InvidiousCourgette](https://avatars.discourse-cdn.com/v4/letter/i/e47c2d/32.png) [@InvidiousCourgette](https://boards.straightdope.com/u/InvidiousCourgette)\ **Post date:** [July 23, 2004, 10:13pm UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585/3 "2004-07-23T22:13:15Z")
yes, the dots are for multiplication. ---
**Author:** ![TJdude825](https://avatars.discourse-cdn.com/v4/letter/t/dec6dc/32.png) [@TJdude825](https://boards.straightdope.com/u/TJdude825)\ **Post date:** [July 23, 2004, 10:24pm UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585/4 "2004-07-23T22:24:39Z")
I don’t know the answer, but this should make it easier to read: (x + ½)[sup](x + ½)[/sup] = (k)(x!)(_e_[sup]x[/sup]) or k = (x + ½)[sup](x + ½)[/sup]/[(x!)(_e_[sup]x[/sup])] ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [July 23, 2004, 10:54pm UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585/5 "2004-07-23T22:54:03Z")
> [@InvidiousCourgette](#): > > If you consider the formula: > > (x+1/2)^(x+1/2) = k. x!.e^x > > As x gets large k quickly moves towards a value of around 0.658 something. Is it possible to say what value k takes as x tends to infinity? Is it a special number, like pi/4 or what have you? Stirling’s formula has the usual form: x! ~ x^(x+1/2) e^(-x) (2 pi)^(1/2) where the tilde (~) indicates an asymptotic relationship. So if you plot the function y(x) = x! e^(x) / x^(x + 1/2) it will approach sqrt(2 pi) as x increases without bound. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [July 23, 2004, 11:37pm UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585/6 "2004-07-23T23:37:14Z")
**biqu** has the right approach using Stirling’s formula, but I believe s/he has a mistake in the limit calculation. I’m getting **sqrt[e/(2pi)]**. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [July 24, 2004, 2:36am UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585/7 "2004-07-24T02:36:39Z")
> [@Cabbage](#): > > **biqu** has the right approach using Stirling’s formula, but I believe s/he has a mistake in the limit calculation. I’m getting **sqrt[e/(2pi)]**. The local version of Maple agrees with **biqu** : it gives me sqrt(2\*pi) as well. ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [July 24, 2004, 2:38am UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585/8 "2004-07-24T02:38:00Z")
> [@Cabbage](#): > > **biqu** has the right approach using Stirling’s formula, but I believe s/he has a mistake in the limit calculation. I’m getting sqrt[e/(2pi)]. Yes, that’s precisely the limit that answers the OP. The limit I gave was for the ratio x! e^(x) / x^(x + 1/2), which follows directly from Stirling’s formula without additional manipulation of limits with logarithms and L’Hospital’s rule. When I went back and applied those techniques to the ratio in the OP, the limit quoted by **Cabbage** fell out immediately. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [July 24, 2004, 2:39am UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585/9 "2004-07-24T02:39:56Z")
I just plain goofed up. Ignore me and listen to **biqu** and **Cabbage**. I’ll just slink away now. ---
**Author:** ![InvidiousCourgette](https://avatars.discourse-cdn.com/v4/letter/i/e47c2d/32.png) [@InvidiousCourgette](https://boards.straightdope.com/u/InvidiousCourgette)\ **Post date:** [July 24, 2004, 6:23am UTC](https://boards.straightdope.com/t/maths-dopers-question-for-you/256585/10 "2004-07-24T06:23:57Z")
🙂 Thanks for the very good answers.