in the previous reply we saw something interesting about f[10^10^x], f[10^10^x] being isomorphic to 10^x Log[10]/2: as x grows without bound f[10^10^x]*10^10^x == 10^x Log[10]/2, but those arguments are so big and the results are so small, it is hard to conceptualize! I just now stumbled on to something interesting about the easier to imagine f[10^x] : As x grows without bound f[10^x] == 10^-x x Log[10]/2 . The former is just a special case of the later.. As always, your comments are welcome!
f[x_] := NSum[(-1)^n (n^(1/n) - 1), {n, x, Infinity},
Method -> "AlternatingSigns", WorkingPrecision -> 400]
You can see as x grows without bound f[10^x] == 10^-x x Log[10]/2 in the following two pieces of code and results:
First we will compute some values of f:
In[126]:= Table[N[f[10^(x)], 30], {x, 0, 20}]
Out[126]= {0.187859642462067120248517934054, \
0.133553295001578355542577811018, -0.976341352833974415772015902634, \
0.00346732160172991953948973985134, \
0.000460749704412491717791977788008, \
0.0000575682039917382492456392183551,
6.90780620020678040526552496824*10^-6,
8.05905469826083516701897111654*10^-7,
9.21034126383164238279161762724*10^-8,
1.03616330307674596183280961166*10^-8,
1.15129254787756199809757665899*10^-9,
1.26642180131318965307268175465*10^-10,
1.38155105581618001913660951164*10^-11,
1.49668031044844208057889632681*10^-12,
1.61180956509609958086783296937*10^-13,
1.72693881974556492465977490514*10^-14,
1.84206807439524003003258758969*10^-15,
1.95719732904493922401341786377*10^-16,
2.07232658369464115957273029594*10^-17,
2.18745583834434340470892771857*10^-18,
2.30258509299404568455944419120*10^-19}
Then look the the difference between f[10^x] and x/2 Log[10] 10^-x. Notice the negative magnitude is doubled already, as x gets to 20.
In[133]:= Table[N[f[10^x] - x/2 Log[10] 10^-x, 30], {x, 0, 20}]
Out[133]= {0.187859642462067120248517934054, \
0.0184240403518760713416782382842, -0.999367203763914872612195817181, \
0.0000134439622388510134627526693130,
2.32685813682580988379497071386*10^-7,
3.57666688710714518943198794863*10^-9,
5.09212246433532115506041838758*10^-11,
6.87278167527295600102514565496*10^-13,
8.91855459646719651808498865426*10^-15,
1.12294254040247134570557355366*10^-16,
1.38053915608858093164841550814*10^-18,
1.66464526862786454571667564300*10^-20,
1.97526087258146388312728449837*10^-22,
2.31238596720188126042014063305*10^-24,
2.67602055238951095541466771904*10^-26,
3.06616462813141282526360313271*10^-28,
3.48281819442594105125308186257*10^-30,
3.92598125127289004134797998316*10^-32,
4.39565379867223450785790828180*10^-34,
4.89183583662397138196824776735*10^-36,
5.41452736512810029562830508680*10^-38}
Any comments as to why this is, would be welcome!