Part 14

1.036927755143369926331365486457034168057080919501912811974192677 9038035897862814845600431065571333363796203414665566090428009617 7915597084183511072180087644866286337180353598363962365128888981 3352767752398275032022436845766444665958115993917977745039244643 9196666159664016205325205021519226713512567859748692860197447984 3200672681297530919900774656558601526573730037561532683149897971 9350398378581319922884886425335104251602510849904346402941172432 7576341508162332245618649927144272264614113007580868316916497918

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Zeta(7) to 512 places : sum(1/n**7,n=1..infinity);

1.008349277381922826839797549849796759599863560565238706417283136 5716014783173557353460969689138513239689614536514910748872867774 1984033544031579830103398456212106946358524390658335396467699756 7696691427804314333947495215378902800259045551979353108370084210 7329399046107085641235605890622599776098694754076320000481632951 2586769250630734413632555601360305007373302413187037951026624779 3954650225467042015510405582224239250510868837727077426002177100 0195455778989836046745406121952650765461161356548679150080858554

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Zeta(9) or sum(1/n**9,n=1..infinity);

1.002008392826082214417852769232412060485605851394888756548596615 9097850533902583989503930691271695861574086047658470602614253739 7072243015306913249876425109092948687676545396979415407826022964 1544836250668629056707364521601531424421326337598815558052591454 0848901539527747456133451028740613274660692763390016294270864220 1123162209241265753326205462293215454665179945038662778223564776 1660330281492364570399301119383985017167926002064923069795850945 8457966548540026945118759481561430375776154443343398399851419383

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This number, the Product[Cos[Pi/n], {n,3,infinity}]

is the limit of an interesting figure in geometry.: If we take a circle, inscribe a triangle, then incribe another circle inside the triangle, then inscribe a square inside the inner circle, then inscribe another circle inside the square, then inscribe a pentagon...

The radius of this figure (the number of sides of the polygon increase with every step:triangle 3, square 4, pentagon 5, ...) approaches a limit: Product[Cos[Pi/n], {n,3,infinity}] Is there any way to get an analytic solution to this? Like this would be the square root of Pi or some combination of radicals and irrational numbers? Anyway, Thanks, Mounitra Chatterji mounitra@seas.ucla.edu

mentioned in december 1995. By Mounitra Chatterji

.1149420448532962007010401574695987428307953372008635168440233965;

maple routine --> product(cos(Pi/n),n=3..infinity);evalf(",64);

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The request was sent by achim flammenkamp on Tue Feb 27 09:05:13 PST 1996 The email address is: achim@mathematik.uni-jena.de The number is 1.60140224354988761393325 (to 24 digits of precision).

-int(sqrt(x)/log(1-x),x=0..1);

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.283265121310307732587685540450858868452123075913479495609303244760289207466703551200728343246718266 1721794706326872389237418265273196389116929121819750888062495294277256191719424273967384545908106616 5124702322513598413388920213387535350692362866707758376138858482266928332718882186473891252470626193 1134162075403008037881499615240658150936661712754874529120769279078826146925069339158824377250780006 81691683658433538480533518043146405030754456294577975558177142447872562829157

There is a pattern in the binary expansion of this number.

The request was sent by B.J. Mares on Sun Dec 3 15:20:18 PST 1995 The email address is: bjmares@teleport.com

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The request was sent by Joe Keane on Sun Sep 10 05:02:26 PDT 1995 The email address is: jgk@netcom.com

The number to be tested is: 1.38432969165678691636600070469187275993602894672280031682863878069088210808356345

The number of correct digits in the number: 79

The hints given by the user:

It's log((3+sqrt(7))/sqrt(2)) or 1/2*arccosh(8).

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