In[290]:= Integrate[Cos@x/(x^2+1),{x,-∞,∞}] Out[290]= π/*e* In[291]:= ContinuedFraction[%,105] Out[291]= {1,6,2,2,1,2,6,8,2,1,1,1,4,3,1,1,66,2,1,1,2,4,2,7,46,10,2,1,18,1,23,10,14,1,1,4,3,1,6,2,5,1,1,1,18,355, 1,1,5,3,4,1,28,5,2,1,3,16,3,25,4,7,1,11,2,2,1,9,2,31,1,20,1,2,1,1,6,13,1,3,1,3,4,125,1,7,2,2,1,3,5,1,1,6,1,1,1,9,1,1,4,6,4,1,13} (Prove it's endless; Win a Fields Medal.) In[292]:= {GeometricMean@%} (*In[295]:= N@Khinchin Out[295]= 2.68545200106531*) Out[292]= {2^(8/15) 3^(23/105) 5^(13/105) 7^(1/21) 2201^(1/105) 3289^(2/105)} In[293]:= NestList[(Print[N@#];GeometricMean@ContinuedFraction [#,105])&,%[[-1]],9] During evaluation of In[293]:= 3.09506434681568 During evaluation of In[293]:= 2.41520362449092 During evaluation of In[293]:= 2.9335577403418 During evaluation of In[293]:= 2.44930689361668 During evaluation of In[293]:= 2.71218585275826 During evaluation of In[293]:= 2.6450717647845 During evaluation of In[293]:= 2.77752000086026 During evaluation of In[293]:= 2.74022410644692 During evaluation of In[293]:= 2.59352990789751 Out[293]= {2^(8/15) 3^(23/105) 5^(13/105) 7^(1/21) 2201^(1/105) 3289^(2/105),2^(18/35) 3^(29/105) 5^(1/35) 104977873^(1/105),2^(19/35) 3^(22/105) 5^(3/35) 7^(2/21) 11^(2/105) 39497^(1/105),2^(7/15) 3^(4/21) 5^(8/105) 7^(2/35) 11^(1/35) 589^(1/105),2^(7/15) 3^(2/7) 5^(8/105) 7^(2/35) 19^(2/105) 1639^(1/105),2^(64/105) 3^(5/21) 5^(8/105) 209^(2/105) 851^(1/105),2^(53/105) 3^(4/21) 5^(11/105) 13^(4/105) 77^(1/35) 1919^(1/105),2^(47/105) 3^(6/35) 5^(11/105) 11^(1/35) 91^(2/105) 316238983^(1/105),2^(3/5) 3^(8/35) 5^(2/35) 7^(4/105) 11^(1/35) 221^(1/105),2^(43/105) 3^(13/105) 35^(8/105) 242280181^(1/105)} In[294]:= NestList[(Print[N@#];GeometricMean@ContinuedFraction [#,105])&,%[[-1]],42] During evaluation of In[294]:= 2.39797710382867 During evaluation of In[294]:= 2.69893228647732 During evaluation of In[294]:= 3.29061256432359 During evaluation of In[294]:= 2.03162608678672 During evaluation of In[294]:= 3.01906752481969 During evaluation of In[294]:= 2.59538767809429 During evaluation of In[294]:= 2.68778297494002 During evaluation of In[294]:= 3.11804540780474 During evaluation of In[294]:= 2.98027331397385 During evaluation of In[294]:= 2.23842419582078 During evaluation of In[294]:= 2.73195233677031 During evaluation of In[294]:= 2.32112558219677 During evaluation of In[294]:= 2.65266653473585 During evaluation of In[294]:= 2.45995266400843 During evaluation of In[294]:= 1.94020851734991 During evaluation of In[294]:= 2.13798184316968 During evaluation of In[294]:= 2.75100265543326 During evaluation of In[294]:= 3.56350709479461 During evaluation of In[294]:= 2.59571502601519 During evaluation of In[294]:= 2.4740594814946 During evaluation of In[294]:= 2.38638422095896 During evaluation of In[294]:= 2.94605839370828 During evaluation of In[294]:= 2.82204016953752 During evaluation of In[294]:= 2.25028325915228 During evaluation of In[294]:= 2.36248747987146 During evaluation of In[294]:= 2.39582513918321 During evaluation of In[294]:= 2.79821884430297 During evaluation of In[294]:= 2.51021506839255 During evaluation of In[294]:= 2.52467515174716 During evaluation of In[294]:= 2.40158175513891 During evaluation of In[294]:= 2.54575104382523 During evaluation of In[294]:= 2.70499590456546 During evaluation of In[294]:= 2.36456538650387 During evaluation of In[294]:= 2.82338870219823 During evaluation of In[294]:= 2.33636891461969 During evaluation of In[294]:= 2.4610503767373 During evaluation of In[294]:= 2.50128322134356 During evaluation of In[294]:= 2.43209873810393 During evaluation of In[294]:= 2.83712666091056 During evaluation of In[294]:= 2.92217685976039 During evaluation of In[294]:= 2.58135001107047 During evaluation of In[294]:= 2.97981675199117 Out[294]= {2^(43/105) 3^(13/105) 35^(8/105) 242280181^(1/105),2^(7/15) 3^(16/105) 5^(1/21) 77^(1/35) 54503796054221^(1/105),2^(58/105) 3^(41/105) 5^(1/15) 17^(2/105) 77^(4/105) 247^(1/105),2^(43/105) 3^(17/105) 5^(2/35) 133^(2/105) 671^(1/105),2^(11/21) 3^(4/21) 5^(8/105) 7^(2/35) 17^(1/35) 437^(1/105) 4433^(2/105),2^(58/105) 3^(29/105) 13^(4/105) 17^(1/35) 11165^(1/105),2^(7/15) 3^(19/105) 5^(3/35) 19^(1/35) 2639^(2/105) 19987^(1/105),2^(3/5) 3^(8/35) 17^(1/35) 35^(1/21) 209^(2/105) 245401^(1/105),2^(58/105) 3^(4/15) 11^(1/35) 35^(4/105) 1417^(2/105) 2369^(1/105),2^(41/105) 3^(16/105) 5^(1/15) 7^(2/105) 11^(1/35) 11497861^(1/105),2^(46/105) 3^(4/15) 5^(13/105) 7^(4/105) 1431859^(1/105),2^(17/35) 3^(1/5) 5^(1/21) 91^(2/105) 410533^(1/105),2^(58/105) 3^(19/105) 11^(1/35) 35^(2/35) 374153^(1/105),2^(59/105) 3^(23/105) 35^(1/21) 39121^(1/105),2^(18/35) 3^(2/21) 5^(2/35) 17^(1/105) 77^(2/105),2^(13/35) 3^(4/21) 5^(8/105) 11^(1/35) 44863^(1/105),2^(19/35) 3^(8/35) 5^(2/35) 7^(4/105) 11^(2/105) 13^(1/35) 34523^(1/105),2^(22/35) 3^(8/35) 5^(1/15) 7^(2/35) 17^(1/35) 143^(2/105) 460112447^(1/105),2^(31/105) 3^(32/105) 5^(13/105) 7^(1/21) 11^(1/35) 17^(2/105),2^(11/21) 3^(19/105) 5^(1/15) 19^(2/105) 77^(1/35) 377^(1/105),2^(41/105) 3^(9/35) 5^(4/105) 17^(1/35) 77^(2/105) 15067^(1/105),2^(22/35) 3^(26/105) 5^(1/15) 7^(4/105) 11^(2/35) 299^(1/105),2^(18/35) 3^(23/105) 5^(13/105) 17^(1/35) 77^(2/105) 3379^(1/105),2^(3/7) 3^(23/105) 5^(1/21) 7^(1/15) 1133^(1/105),2^(16/35) 3^(6/35) 11^(1/35) 13^(2/105) 35^(1/21) 1241^(1/105),2^(3/7) 3^(4/21) 5^(1/21) 11^(1/35) 13^(2/105) 80741659^(1/105),2^(3/7) 3^(16/105) 31^(2/105) 35^(1/15) 221^(1/35) 82777^(1/105),2^(8/21) 3^(9/35) 5^(1/21) 11^(1/35) 26739058073^(1/105),2^(16/35) 3^(5/21) 7^(1/15) 11^(1/35) 115^(2/105) 493^(1/105),2^(18/35) 3^(23/105) 5^(1/21) 7^(1/35) 19^(2/105) 13651^(1/105),2^(7/15) 3^(23/105) 11^(1/35) 35^(1/15) 899^(1/105),2^(62/105) 3^(19/105) 5^(4/105) 7^(1/21) 247^(2/105) 694331^(1/105),2^(41/105) 3^(2/15) 5^(2/21) 7^(1/21) 13^(4/105) 35563^(1/105),2^(18/35) 3^(6/35) 5^(8/105) 7^(2/105) 377^(1/35) 29932163^(1/105),2^(37/105) 3^(19/105) 5^(1/15) 7^(2/35) 11^(2/105) 2800733^(1/105),2^(41/105) 3^(1/7) 5^(3/35) 7^(1/21) 112819455463^(1/105),2^(10/21) 3^(6/35) 5^(1/15) 7^(1/35) 221^(2/105) 1123441^(1/105),2^(8/15) 3^(1/7) 5^(1/15) 7^(1/21) 24845483^(1/105),2^(53/105) 3^(26/105) 5^(8/105) 7^(2/35) 338981071^(1/105),2^(47/105) 3^(29/105) 5^(4/35) 7^(3/35) 13^(2/105) 493^(1/105),2^(46/105) 3^(19/105) 5^(8/105) 77^(1/35) 1202287073^(1/105),2^(64/105) 3^(5/21) 7^(4/105) 55^(1/15) 1079^(1/105),2^(61/105) 3^(6/35) 5^(1/15) 7^(1/21) 13^(2/105) 3023603^(1/105)} Methinks this could have a largish period. Limit? LUB? GLB? Tighter for larger values of "105"? —rwg