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 Infinite Product (Posted on 2013-06-24)
Find the value of the infinite product
P=7/9*26/28*63/65*....*(k3-1)/(k3+1)*...

 No Solution Yet Submitted by Danish Ahmed Khan No Rating

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 computer exploration (spoiler) | Comment 2 of 3 |

5   P=1
10   for K=2 to 40
20      P=P*(K^3-1)/(K^3+1)
30      print K,P
40   next

finds the partial products up to various k:

`max     partial product k2       0.77777777777777777773       0.72222222222222222214       0.75       0.68888888888888888876       0.68253968253968253957       0.67857142857142857128       0.67592592592592592579       0.674074074074074073810      0.67272727272727272711      0.671717171717171716812      0.670940170940170939813      0.670329670329670329214      0.669841269841269840715      0.669444444444444443916      0.669117647058823528817      0.668845315904139432918      0.66861598440545808919      0.668421052631578946720      0.668253968253968253221      0.668109668109668108922      0.667984189723320157323      0.667874396135265699724      0.667777777777777776925      0.667692307692307691526      0.667616334283000948827      0.667548500881834214328      0.667487684729064038529      0.667432950191570880330      0.667383512544802866431      0.667338709677419353932      0.66729797979797979733      0.667260843731431965834      0.6672268907563025235      0.667195767195767194736      0.667167167167167166137      0.667140825035561876638      0.667116509221772378539      0.667094017094017092840      0.6670731707317073159`

and, when extended there seems to be a pattern in the partial products of max k = 4 x 10^i (more precision was used here than shown, to allow for losses in rounding at each step):

`40              0.6670731707317073170731707400             0.66667082294264339152119704000            0.666666708322919270182454340000           0.6666666670833229169270768400000          0.66666666667083332291669274000000         0.666666666666708333322916640000000        0.6666666666666670833333229`

In any event, the limit seems to be 2/3.

 Posted by Charlie on 2013-06-24 12:29:29

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