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Searching for an asymptotic to exp[0.5]
#7
Code:
a_1= 1.289368074687
a_2= 2.685542084449
a_3= 4.481892104368
a_4= 6.478743767633
a_5= 8.617330003873
a_6= 10.86883764614
a_7= 13.21549097355
a_8= 15.64480318533
a_9= 18.14765340676
a_10= 20.71664375389
a_11= 23.34605172725
a_12= 26.03107295368
a_13= 28.76759185629
a_14= 31.55218923626
a_15= 34.38183207053
a_16= 37.25395409452
a_17= 40.16617725981
a_18= 43.11650203072
a_19= 46.10306787913
a_20= 49.12423329725
a_21= 52.17844415943
a_22= 55.26438661508
a_23= 58.38072977033
a_24= 61.52640919181
a_25= 64.70028943211
a_26= 67.90141159249
a_27= 71.12888751647
a_28= 74.38183008946
a_29= 77.65944293902
a_30= 80.96100674887
a_31= 84.28582227410
a_32= 87.63323120070
a_33= 91.00261369743
a_34= 94.39338557884
a_35= 97.80499516680
a_36= 101.2369199152
a_37= 104.6886902658
a_38= 108.1598101190
a_39= 111.6498571303
a_40= 115.1583810162
a_41= 118.6850011605
a_42= 122.2293283987
a_43= 125.7909699804
a_44= 129.3696195111
a_45= 132.9648897794
a_46= 136.5764573911
a_47= 140.2040557334
a_48= 143.8473645074
a_49= 147.5060942647
a_50= 151.1799685187
a_51= 154.8686996662
a_52= 158.5720762206
a_53= 162.2898267064
a_54= 166.0217152588
a_55= 169.7675306959
a_56= 173.5270093037
a_57= 177.2999786775
a_58= 181.0862092231
a_59= 184.8855000278
a_60= 188.6976766463
a_61= 192.5225306431
a_62= 196.3598778123
a_63= 200.2095425885
a_64= 204.0713710077
a_65= 207.9451779799
a_66= 211.8308191885
a_67= 215.7281180480
a_68= 219.6369408364
a_69= 223.5570997849
a_70= 227.4885071866
a_71= 231.4309843808
a_72= 235.3843931661
a_73= 239.3486192792
a_74= 243.3235140576
a_75= 247.3089677325
a_76= 251.3048375095
a_77= 255.3110228135
a_78= 259.3273875110
a_79= 263.3538324564
a_80= 267.3902471867
a_81= 271.4365024525
a_82= 275.4924928105
a_83= 279.5581471603
a_84= 283.6333119804
a_85= 287.7179392030
a_86= 291.8118947335
a_87= 295.9151023881
a_88= 300.0274504396
a_89= 304.1488634944
a_90= 308.2792519888
a_91= 312.4185117624
a_92= 316.5665730911
a_93= 320.7233354589
a_94= 324.8887317451
a_95= 329.0626821729
a_96= 333.2451078801
a_97= 337.4359118041
a_98= 341.6350372412
a_99= 345.8424068745
a_100= 350.0579336860
a_101= 354.2815588969
a_102= 358.5132133215
a_103= 362.7528106159
a_104= 367.0003160828
a_105= 371.2556293444
a_106= 375.5186854508
a_107= 379.7894500858
a_108= 384.0678314842
a_109= 388.3537801607
a_110= 392.6472189586
a_111= 396.9481223641
a_112= 401.2563957391
a_113= 405.5719986392
a_114= 409.8948585083
a_115= 414.2249480026
a_116= 418.5621839981
a_117= 422.9065248225
a_118= 427.2579174535
a_119= 431.6162930147
a_120= 435.9816335167
a_121= 440.3538537394
a_122= 444.7329047543
a_123= 449.1187668605
a_124= 453.5113755327
a_125= 457.9106702550
a_126= 462.3166165089
a_127= 466.7291677505
a_128= 471.1482630381
a_129= 475.5738880517
a_130= 480.0059669786
a_131= 484.4444725900
a_132= 488.8893596420
a_133= 493.3405854525
a_134= 497.7981092466
a_135= 502.2618743290
a_136= 506.7318543022
a_137= 511.2080081242
a_138= 515.6902967113
a_139= 520.1786803772
a_140= 524.6731047065
a_141= 529.1735644829
a_142= 533.6799885996
a_143= 538.1923555703
a_144= 542.7106288953
a_145= 547.2347567376
a_146= 551.7647258028
a_147= 556.3004650155
a_148= 560.8419393211
a_149= 565.3890426629
a_150= 569.9416519995

It seems 0.5 n (ln(n)-1) < a_n < 2 n (ln(n)-1).

Hence the new conjecture is 1/(n ln(n)) ! as Taylor coefficients.

regards

tommy1729
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Messages In This Thread
RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/10/2014, 11:56 PM

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