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What are the best bars in Paris?
Really...too many to mention I am afraid but let's give it a go. I have identified two categories of bars I quite like and hopefully more Quorans can add to this question and we can create a good Answer Wiki with a Google Map :)First my favorite Cocktail bars, all of them sensational, some of them only "word-of-mouth/no sign on the door/need to book in advance":Ballroom of the Beef Club http://www.eccbeefclub.com/La Candelaria http://www.candelariaparis.com/La Conserverie http://www.laconserveriebar.com/Harry's New York Bar http://www.harrysbar.fr/en/homepage/Jefrey's http://www.jefreys.fr/Acte 3 https://www.facebook.com/pages/Acte-3/138981316209917?sk=infoLes Jalles http://www.lesjalles.fr/Le Magnifique http://www.lemagnifique.fr/Bar 8 - Hotel Mandarin Oriental http://www.mandarinoriental.com/paris/dining/bar_8/L'Entrée des Artistes https://plus.google.com/114118906145169800786/about?gl=US&hl=enLe Bar - Hotel Shangri-La http://www.shangri-la.com/paris/shangrila/dining/bars-lounges/le-bar/Café Marly http://www.maisonthierrycostes.com/cafe-marly/accueilAt the other end of the spectrum are the "underground" or alternative bars, most do not have a website, all of them are very cheap and with a great atmosphere:Aux Folies 8 rue de Belleville 20eBar Ourcq 68 quai de Loire 19eLe Zéro Zéro 89 rue Amelot, 11eLa Fourmi 74 rue des Martyrs 18eBarbershop 68 avenue de la République, 11eLe Dix 10 rue de l'Odéon 6eLe Point Ephémère http://www.pointephemere.org/Ô Paris 1 rue des Envierges 20eLa Mécanique Ondulatoire 8 passage Thiéré 11eLes Combustibles 14 rue Abel, 12e, ParisLe Zorba 137 rue du Faubourg du Temple, 10eKarambole Café 10 rue Hégésippe Moreau, 18e, Paris
How do you evaluate [math]\lim_{x\to\infty} x (1+1/x)^x - ex[/math] using any applicable method?
Instead of solving this specific case, let’s generalize this problem by understanding how [math](1+1/x)^x[/math] behaves when [math]x[/math] is large.The trick is to define [math]p = \frac{1}{x}[/math] and consider [math]p \approx 0[/math]And then finding the Taylor series:[math]\displaystyle (1+1/x)^x = (1+p)^{1/p} = e - \frac{e}{2}p [/math][math][/math][math]+ \frac{11e}{24}p^2 - \frac{7e}{16}p^3 [/math][math][/math][math]+ \frac{2447e}{5760}p^4 - [/math][math][/math][math]...[/math]Reverting back to large values of [math]x[/math], we get:[math]\displaystyle (1+1/x)^x = e - \frac{e}{2x} [/math][math][/math][math]+ \frac{11e}{24x^2} - \frac{7e}{16x^3} [/math][math][/math][math]+ \frac{2447e}{5760x^4} - [/math][math][/math][math]...[/math]With that in hand, we can use Big O notation for asymptotically large [math]x[/math], to write:[math]\displaystyle (1+1/x)^x = e [/math][math][/math][math]+ O(1/x) \implies [/math][math][/math][math][/math][math]\displaystyle [/math][math][/math][math]\ [/math][math][/math][math]\ [/math][math][/math][math]\ \boxed{\lim_{x \to \infty} (1+1/x)^x = e}\ [/math][math][/math][math]\ [/math][math][/math][math]\ [/math][math][/math][math][/math] (this we learn in high-school)[math]\displaystyle x(1+1/x)^x = xe - \frac{e}{2} [/math][math][/math][math]+ O(1/x) \implies[/math][math]\displaystyle [/math][math][/math][math]\ [/math][math][/math][math]\ [/math][math][/math][math]\ \boxed{\lim_{x \to \infty} x(1+1/x)^x - xe = -\frac{e}{2}}\ [/math][math][/math][math]\ [/math][math][/math][math][/math] (this is what the OP wanted to know)[math]\displaystyle x^2(1+1/x)^x = x^2e - \frac{ex}{2} [/math][math][/math][math]+ \frac{11e}{24} [/math][math][/math][math]+ O(1/x) \implies[/math][math]\displaystyle [/math][math][/math][math]\ [/math][math][/math][math]\ [/math][math][/math][math]\ \boxed{\lim_{x \to \infty} x^2(1+1/x)^x - x^2e [/math][math][/math][math]+ \frac{ex}{2} = \frac{11e}{24}}\ [/math][math][/math][math][/math] (this is the next level)[math]\displaystyle x^3(1+1/x)^x = x^3e - \frac{ex^2}{2} [/math][math][/math][math]+ \frac{11ex}{24} - \frac{7e}{16} [/math][math][/math][math]+ O(1/x) \implies[/math][math]\displaystyle [/math][math][/math][math]\ [/math][math][/math][math]\ [/math][math][/math][math]\ \boxed{\lim_{x \to \infty} x^3(1+1/x)^x - x^3e [/math][math][/math][math]+ \frac{ex^2}{2} - \frac{11ex}{24} = -\frac{7e}{16}}\ [/math][math][/math][math][/math] (this is the next next level)And so on…
What was the casualty rate for American tanks and crews in Vietnam?
ARMORED CAV M48A3 TANK CREWMAN VIETNAM 11 ECHOUS ARMY StatisticsUS Army Tankers In 11th ACR KIA in Vietnam = 725 (equals 27% of all tankers ever assigned to Vietnam)The highest loss-rate for any MOS was 11E (Armor Crewman) 27% KIASource: Combat Area Casualty File of 11/93, National ArchivesIn our company alone D Co 1/11 ACR we had 28 deaths. Mostly tankers by fragmentation hits.
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