Here is a twist with an interesting property, where previously

Now let

Some experimenting shows, as x grows without bounds, using only the mantissas, (m-less[10x])/(m-less[x]) quickly converges to Exp[Sin[2/(10^x - 1)]]

m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}, 
   Method -> "AlternatingSigns", WorkingPrecision -> 2000];


standardplus[
  x_] := (NSum[(-1)^n (n^(1/n) - 1), {n, 1, 100^x}, 
   Method -> "AlternatingSigns", WorkingPrecision -> 2000])

less[x_] := (standardplus[x] - 10^-x x Log[10]/2)

Table[a = m - less[10 x]; b = m - less[x];
 Print["(m-less[", 10 x, "])/(m-less[", x, "]=", 
  c = N[MantissaExponent[a][[1]]/MantissaExponent[b][[1]], 80]]; 
 Print[]; Print["(m-less[", 10 x, "])/(m-less[", x, 
  "]-Exp[Sin[2/(10^", x, "-1)]])=", 
  N[c - Exp[Sin[2/(10^x - 1)]], 20]]; Print[];, {x, 2, 25}]

Remember it only uses the Mantissa of of less[n]. Then we get the following.

(m-less[20])/(m-less[2]=1.0204168273548164923933498640032520751266328775998462434645705193180301888879476



(m-less[20])/(m-less[2]-Exp[Sin[2/(10^2-1)]])=0.000010767387677440344049



(m-less[30])/(m-less[3]=1.0020040209556828541303021951774030224057933218149664141378894930634440070383666



(m-less[30])/(m-less[3]-Exp[Sin[2/(10^3-1)]])=1.4949683006455269781*10^-8



(m-less[40])/(m-less[4]=1.0002000400254835609163846503172225148130032358901133160723933568663264622861750



(m-less[40])/(m-less[4]-Exp[Sin[2/(10^4-1)]])=1.9482960917736922803*10^-11



(m-less[50])/(m-less[5]=1.0000200004000300703231945013284774047306162005279327074505030985009631876758722



(m-less[50])/(m-less[5]-Exp[Sin[2/(10^5-1)]])=2.4070263194514680745*10^-14



(m-less[60])/(m-less[6]=1.0000020000040000346673349921829793851323605712210071456875787041476795678356858



(m-less[60])/(m-less[6]-Exp[Sin[2/(10^6-1)]])=2.8667328992183112737*10^-17



(m-less[70])/(m-less[7]=1.0000002000000400000392672264432304585700399350360636614309875891025684888693308



(m-less[70])/(m-less[7]-Exp[Sin[2/(10^7-1)]])=3.3267225843230459903*10^-20



(m-less[80])/(m-less[8]=1.0000000200000004000000438685064877639678054011897844032446586723321998905088372



(m-less[80])/(m-less[8]-Exp[Sin[2/(10^8-1)]])=3.7868506427763967819*10^-23



(m-less[90])/(m-less[9]=1.0000000020000000040000000484706593229923072663272999043578716326640675060479852



(m-less[90])/(m-less[9]-Exp[Sin[2/(10^9-1)]])=4.2470659316992307266*10^-26



(m-less[100])/(m-less[10]=1.0000000002000000000400000000530734166036054429176803065066316870530299919715231



(m-less[100])/(m-less[10]-Exp[Sin[2/(10^10-1)]])=4.7073416603005442918*10^-29



(m-less[110])/(m-less[11]=1.0000000000200000000004000000000576766127062843991852342856967381331227121396773



(m-less[110])/(m-less[11]-Exp[Sin[2/(10^11-1)]])=5.1676612706224399185*10^-32



(m-less[120])/(m-less[12]=1.0000000000020000000000040000000000622801378355028524605933179423989470628700972



(m-less[120])/(m-less[12]-Exp[Sin[2/(10^12-1)]])=5.6280137835496852461*10^-35



(m-less[130])/(m-less[13]=1.0000000000002000000000000400000000000668839160517896971904135458035429418120714



(m-less[130])/(m-less[13]-Exp[Sin[2/(10^13-1)]])=6.0883916051789097190*10^-38



(m-less[140])/(m-less[14]=1.0000000000000200000000000004000000000000714878931210468090906447144542438631803



(m-less[140])/(m-less[14]-Exp[Sin[2/(10^14-1)]])=6.5487893121046749091*10^-41



(m-less[150])/(m-less[15]=1.00000000000000200000000000000400000000000007609202927255176728291187147944873514



(m-less[150])/(m-less[15]-Exp[Sin[2/(10^15-1)]])=7.0092029272551761283*10^-44



(m-less[160])/(m-less[16]=1.00000000000000020000000000000004000000000000008069629467836896918670769567621762



(m-less[160])/(m-less[16]-Exp[Sin[2/(10^16-1)]])=7.4696294678368968587*10^-47



(m-less[170])/(m-less[17]=1.00000000000000002000000000000000040000000000000008530066652891235803298105087808



(m-less[170])/(m-less[17]-Exp[Sin[2/(10^17-1)]])=7.9300666528912357973*10^-50



(m-less[180])/(m-less[18]=1.00000000000000000200000000000000000400000000000000008990512708339406863014739873



(m-less[180])/(m-less[18]-Exp[Sin[2/(10^18-1)]])=8.3905127083394068624*10^-53



(m-less[190])/(m-less[19]=1.00000000000000000020000000000000000004000000000000000009450966233592908476119415



(m-less[190])/(m-less[19]-Exp[Sin[2/(10^19-1)]])=8.8509662335929084761*10^-56



(m-less[200])/(m-less[20]=1.00000000000000000002000000000000000000040000000000000000009911426108180940866022



(m-less[200])/(m-less[20]-Exp[Sin[2/(10^20-1)]])=9.3114261081809408660*10^-59



(m-less[210])/(m-less[21]=1.00000000000000000000200000000000000000000400000000000000000010371891425055713901



(m-less[210])/(m-less[21]-Exp[Sin[2/(10^21-1)]])=9.771891425055713901*10^-62



(m-less[220])/(m-less[22]=1.00000000000000000000020000000000000000000004000000000000000000010832361442087217



(m-less[220])/(m-less[22]-Exp[Sin[2/(10^22-1)]])=1.0232361442087217*10^-64



(m-less[230])/(m-less[23]=1.00000000000000000000002000000000000000000000040000000000000000000011292835546212



(m-less[230])/(m-less[23]-Exp[Sin[2/(10^23-1)]])=1.0692835546212*10^-67



(m-less[240])/(m-less[24]=1.00000000000000000000000200000000000000000000000400000000000000000000011753313227



(m-less[240])/(m-less[24]-Exp[Sin[2/(10^24-1)]])=1.1153313227*10^-70



(m-less[250])/(m-less[25]=1.00000000000000000000000020000000000000000000000004000000000000000000000012213794



(m-less[250])/(m-less[25]-Exp[Sin[2/(10^25-1)]])=1.161379*10^-73

Some serious experimenting indicate, as x grows without bounds,

y(x)=1/x (m - less[10 x])/(m - less[x] - Exp[Sin[2/(10^x - 1)]])

goes to 0 like approximately 4.60 *10^(-3 x);

see attached "limit1 over x less-expsin.nb," or the following table.

x         y(x)

25       4.6455176215736139*(10^-75)
50       4.6252570448844720*(10^-150)
75       4.6185421232753716*(10^-225)
100     4.61519190071218653*(10^-300)

250      4.60917366034394659*(10^-750)
300      4.60850593206854638*(10^-900)
350     4.608029101475772605*(10^-1050)
400     4.607671543158347315*(10^-1200)
450     4.607393480542367672*(10^-1350)
500   4.6071710545770551745*(10^-1500)

Attachments:

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!