[math-fun] another circuitous derivation
had among its terminal nodes In[652]:= Sum[(-1)^n*ArcTan[Csch[Pi*(n - 1/2)/Sqrt[3]]], {n, \[Infinity]}] == -5*Pi/24 Out[652]= 1 (n - -) Pi n 2 Sum[(-1) ArcTan[Csch[-------------]], n >= 1] Sqrt[3] 5 Pi = - ----. 24 It doesn't *look* that hard ... Joerg? Easy warmup: In[654]:= Product[Coth[Pi*(n-1/2)]^8, {n, \[Infinity]}] == 2 Out[654]= 1 8 Product[Coth[(n - -) Pi] , n >= 1] = 2. 2 --rwg
You either need to make sure the text is all one line or that it's formatted correctly--as written, it doesn't parse. On Sat, Jan 23, 2010 at 9:58 PM, Bill Gosper <billgosper@gmail.com> wrote:
had among its terminal nodes In[652]:= Sum[(-1)^n*ArcTan[Csch[Pi*(n - 1/2)/Sqrt[3]]], {n, \[Infinity]}] == -5*Pi/24 Out[652]= 1 (n - -) Pi n 2 Sum[(-1) ArcTan[Csch[-------------]], n >= 1] Sqrt[3] 5 Pi = - ----. 24 It doesn't *look* that hard ... Joerg? Easy warmup: In[654]:= Product[Coth[Pi*(n-1/2)]^8, {n, \[Infinity]}] == 2 Out[654]= 1 8 Product[Coth[(n - -) Pi] , n >= 1] = 2. 2 --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
Gack. It looks like I can't send formatted ASCII from Gmail. (Squirrelmail has just gotten even more useless.) It looks OK in my Gmail compose window, but when I copy and paste it elsewhere, it is ruined. Can you think of a workaround, or do I need to recompose it from scratch, and drastically curtail my SquirrelMail BCCs? --rwg
You either need to make sure the text is all one line or that it's formatted correctly--as written, it doesn't parse.
On Sat, Jan 23, 2010 at 9:58 PM, Bill Gosper <billgosper@gmail.com> wrote:
had among its terminal nodes In[652]:= Sum[(-1)^n*ArcTan[Csch[Pi*(n - 1/2)/Sqrt[3]]], {n, \[Infinity]}] == -5*Pi/24 Out[652]= 1 (n - -) Pi n 2 Sum[(-1) ArcTan[Csch[-------------]], n >= 1] Sqrt[3] 5 Pi = - ----. 24 It doesn't *look* that hard ... Joerg? Easy warmup: In[654]:= Product[Coth[Pi*(n-1/2)]^8, {n, \[Infinity]}] == 2 Out[654]= 1 8 Product[Coth[(n - -) Pi] , n >= 1] = 2. 2 --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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Gack. It looks like I can't send formatted ASCII from Gmail. (Squirrelmail has just gotten even more useless.) It looks OK in my Gmail compose window, but when I copy and paste it elsewhere, it is ruined. Can you think of a workaround, or do I need to recompose it from scratch, and drastically curtail my SquirrelMail BCCs? --rwg
When I click on plain text on the resend, it destroys my whitespace, and won't let me use Courier font! How plain is that? Here's a partial repaste from Mma: In[7]:= HoldForm[Sum][(-1)^n* ArcTan[Csch[Pi*(n - 1/2)/Sqrt[3]]], {n, \[Infinity]}] == -5*Pi/24 Out[7]= 1 (-(-) + n) Pi n 2 5 Pi Sum[(-1) ArcTan[Csch[-------------]], {n, Infinity}] == -(----) Sqrt[3] 24 In[8]:= OutputForm[Product[Coth[Pi*(n - 1/2)]^8, {n, \[Infinity]}] == 2] Out[8]//OutputForm= 1 8 Product[Coth[(-(-) + n) Pi] , {n, Infinity}] == 2 2 --rwg
You either need to make sure the text is all one line or that it's formatted correctly--as written, it doesn't parse.
On Sat, Jan 23, 2010 at 9:58 PM, Bill Gosper <billgosper@gmail.com> wrote:
had among its terminal nodes In[652]:= Sum[(-1)^n*ArcTan[Csch[Pi*(n - 1/2)/Sqrt[3]]], {n, \[Infinity]}] == -5*Pi/24 Out[652]= 1 (n - -) Pi n 2 Sum[(-1) ArcTan[Csch[-------------]], n >= 1] Sqrt[3] 5 Pi = - ----. 24 It doesn't *look* that hard ... Joerg? Easy warmup: In[654]:= Product[Coth[Pi*(n-1/2)]^8, {n, \[Infinity]}] == 2 Out[654]= 1 8 Product[Coth[(n - -) Pi] , n >= 1] = 2. 2 --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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On Tue, Jan 26, 2010 at 4:10 AM, <rwg@sdf.lonestar.org> wrote:
Gack. It looks like I can't send formatted ASCII from Gmail.
Sure you can. Make sure you're in plain text mode and then paste in your formatted text. If you're reading formatted ascii in gmail, you can either install a greasemonkey script to force it to be fixed-width font all the time, or you can click on the little blue triangle in the upper right-hand corner of the message and select "Show original". The one thing I really dislike about Gmail's support for formatted ascii is that if the first character is a space, it removes it. So you either have to add an extra space to those lines or prepend every line with a non-space character: . ___ _ . / __|___ ___ __ _| |___ .| (_ / _ \/ _ \/ _` | -_) . \___\___/\___/\__, |_\___| |___/ (This logo replaces the Google image when you search for "ascii art". :) -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
The warm-up product is very beautiful; I've never seen it before. I can use the fact that (1/2)*log|(1+x)/(1-x)| = x + x^3/3 + x^5/5 + x^7/7 + ... for |x| < 1 to transform log(coth(pi/2)) + log(coth(3*pi/2) + log(coth(5*pi/2) + ... to csch(pi) + csch(3*pi)/3 + csch(5*pi)/5 + ... . So it remains to prove that the last series is equal to log(2)/8. This probably could be done with residue calculus, but I haven't tried using it to sum anything like this before. So, that's it for my attempt at the "easy" warm-up. Warut On Sun, Jan 24, 2010 at 12:58 PM, Bill Gosper <billgosper@gmail.com> wrote:
had among its terminal nodes In[652]:= Sum[(-1)^n*ArcTan[Csch[Pi*(n - 1/2)/Sqrt[3]]], {n, \[Infinity]}] == -5*Pi/24 Out[652]= 1 (n - -) Pi n 2 Sum[(-1) ArcTan[Csch[-------------]], n >= 1] Sqrt[3] 5 Pi = - ----. 24 It doesn't *look* that hard ... Joerg? Easy warmup: In[654]:= Product[Coth[Pi*(n-1/2)]^8, {n, \[Infinity]}] == 2 Out[654]= 1 8 Product[Coth[(n - -) Pi] , n >= 1] = 2. 2 --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Bill Gosper -
Mike Stay -
rwg＠sdf.lonestar.org -
Warut Roonguthai