Rich just pointed out to me that
product(2*cos(3*t/2^n)+sqrt(7)*sin(t/2^n)-cos(t/2^n),n,1,inf) = (2*sin(3*t)+sin(t))/sqrt(7)+cos(t)
inf /===\ | | 3 t t t 2 sin(3 t) + sin(t) | | (2 cos(---) + sqrt(7) sin(--) - cos(--)) = ------------------- + cos(t) | | n n n sqrt(7) n = 1 2 2 2
is equivalent to prod((x^2^k-%e^(3*%i*%pi/7))*(x^2^k-%e^(5*%i*%pi/7))*(x^2^k+%e^(6*%i*%pi/7)),k,0,n-1) = (x^2^n+%e^(3*%i*%pi/7))*(x^2^n+%e^(5*%i*%pi/7))*(x^2^n-%e^(6*%i*%pi/7))/((x+%e^(3*%i*%pi/7))*(x+%e^(5*%i*%pi/7))*(x-%e^(6*%i*%pi/7))) n - 1 3 i pi 5 i pi 6 i pi /===\ k ------ k ------ k ------ | | 2 7 2 7 2 7 | | (x - e ) (x - e ) (x + e ) = | | k = 0 3 i pi 5 i pi 6 i pi n ------ n ------ n ------ 2 7 2 7 2 7 (x + e ) (x + e ) (x - e ) ----------------------------------------------- 3 i pi 5 i pi 6 i pi ------ ------ ------ 7 7 7 (x + e ) (x + e ) (x - e ) ! Trying to derive Rich's observation in Mma 7.0, In[1]:= Factor[((x*(Sqrt[7]*I - 1))/2) + ((x^2*(-Sqrt[7]*I - 1))/2) + x^3 + 1, Extension -> (-1)^(1/7)] Out[1]= -(I ((-93222968909115529994920 + 20924612493340656725134 I) + (86082923884130960518872 + 357456564149362034379738 I) (-1)^( 1/7) + (279230634648179756081536 - 254577319249669754726972 I) (-1)^( 2/7) - (128909861890973390508704 + 19261800404123111281744 I) (-1)^( 3/7) - (262911491903882522699168 + 1717802482722402797698 I) (-1)^( 4/7) + (45621554544051365632984 + 273756075804862968415245 I) (-1)^( 5/7) - (81719779538053027762566 - 18851209912033826322536 I) Sqrt[ 7] + (103154450766576283430054 + 105678145987231257343688 I) (-1)^(1/7) Sqrt[ 7] + (59292997013846884998120 - 16002776632415384530112 I) (-1)^(2/7) Sqrt[ 7] - (23383123842825850913776 + 81972872431225606590752 I) (-1)^(3/7) Sqrt[ 7] - (87232069186717876328194 - 60789769565100139284672 I) (-1)^(4/7) Sqrt[ 7] + (13861719860507037285844 + 61026548391231898078632 I) (-1)^(5/7) Sqrt[7] - 9489147790175370672293 I x) ((-231095869671025050999320 - 29770809283501669680960 I) + (80599420984539853865720 + 866974483270444659023770 I) (-1)^( 1/7) + (699268986371722210933416 - 527606990314207937182708 I) (-1)^( 2/7) - (253276297158528052883264 + 96224042577750374127857 I) (-1)^( 3/7) - (633197087550431243870744 + 67151907184397843513304 I) (-1)^( 4/7) + (77313257087604747957600 + 638476440045894637928148 I) (-1)^( 5/7) - (190459539326979985517456 - 20473275286207017234552 I) Sqrt[ 7] + (200807251268965307405146 + 255243231270689268830856 I) (-1)^(1/7) Sqrt[ 7] + (161654254346412719028592 - 8037913225998504703144 I) (-1)^(2/7) Sqrt[ 7] - (33708196756104193101296 + 216156877780788618401216 I) (-1)^(3/7) Sqrt[ 7] - (208290422612047683845544 - 118883493814378106221976 I) (-1)^(4/7) Sqrt[ 7] + (19030177672653028736736 + 139410137615409039410592 I) (-1)^(5/7) Sqrt[7] + 9489147790175370672293 I x) ((-247532495673239390006288 - 74063007060472274128561 I) + (3995463777580813664096 + 1096160552562377585772601 I) (-1)^( 1/7) + (920251889666913795020568 - 652529069234524320640525 I) (-1)^( 2/7) - (295647348213202062228832 + 95401146337229889537207 I) (-1)^( 3/7) - (797133471283395593852552 + 152391977716105965086363 I) (-1)^( 4/7) + (74489256912699605780152 + 822156194011111601909433 I) (-1)^( 5/7) - (245298890246438957961048 - 13932194855161190302192 I) Sqrt[ 7] + (242227830288253477199700 + 312940817233212567451072 I) (-1)^(1/7) Sqrt[ 7] + (196234992398398299759540 + 24422733428319569334824 I) (-1)^(2/7) Sqrt[ 7] - (22473251874752216955528 + 290128839578604404880864 I) (-1)^(3/7) Sqrt[ 7] - (264124256011717110056402 - 148905955741245735693992 I) (-1)^(4/7) Sqrt[ 7] + (19312095402862888770284 + 168958679078555069590216 I) (-1)^(5/7) Sqrt[7] + 9489147790175370672293 I x))/ 854440119369967104087688437612993743725942514691531406700822737757 whose individual factors seem beyond its ability to simplify! Macsyma just croaks "Quotient by zero", which is not surprising since there are many nonobvious zeros lurking: 6 %i %pi 5 %i %pi -------- -------- 7 7 (sqrt(7) %i - 1) %e - (sqrt(7) %i + 1) %e (d8) --------------------------------------------------------- 2 4 %i %pi -------- 7 + %e - 1 (c9) EXPAND(DFLOAT(%)); (d9) 1.11022302462516d-16 %i --rwg Drat! The lycopene just took off and crashed from the same incline where the vitamin E hasn't budged.