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Exploring the Upper End of the Partial Sums of the MRB constant in Mma

The MRB constant is defined as the upper limit point of the partial sums of the divergent series, s[k] =Sum[(-1)^n n^(1/n), {n, 1, k}]. It is computed moire efficiently by the convergent summation, CMRB =Sum[(-1)^n (n^(1/n)-1), {n, 1, Infinity}]; In Mathematica, a simple command to compute some digits is, NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 200] .

One must sum a number in the order of 10^k iterations of s1[k] =Sum[(-1)^n (n^(1/n)-1), {n, 1, k}] to get k decimal places of the MRB constant. See following table.

After 10^1 iterations: 0.3

After 10^2 iterations: 0.21

After 10^3 iterations: 0.191

After 10^4 iterations: 0.1883

After 10^5 iterations: 0.18792

After 10^6 iterations: 0.187867

After 10^7 iterations: 0.1878604

After 10^8 iterations: 0.18785973

After 10^9 iterations: 0.187859653

After 10^10 iterations: 0.1878596436

After 10^11 iterations: 0.18785964259

After 10^12 iterations: 0.187859642476

 etc.

After an infinite number of iterations, you arrive at the MRB constant.

But that might be the wrong direction to go to get an exact formula for the MRB constant or determine its irrationality!

Lets look at the last several partial sums of s1[k] =Sum[(-1)^n (n^(1/n)-1), {n, 1, k}] : The upper end of partial sums here are more easily computed than are the ones of s[k] =Sum[(-1)^n n^(1/n), {n, 1, k}].. I might cover those later.

First lets define f to give several digits from the upper end of the partial sums of the MRB constant :

f[x_] := NSum[(-1)^n (n^(1/n) - 1), {n, x, Infinity},  Method -> "AlternatingSigns", WorkingPrecision -> 500]

Some exploring shows that when x grows without bounds, for integer a,f[10^(x + a)] == f[10^x]/(10^a x/(x + a))

Here are some partial examples:

Table[N[f[10^x]/f[10^(x + 1)], 30] - 10^1 x/(x + 1), {x, 1, 398}]

giving

{0.64500979554588672227870453856, 0.15665400662241862356341215692, \
0.02539083264556616595410384804, 0.00354488179994308588728668282, \
0.00045634218115704700459279609, 0.00005587537323876003255331404, 
 6.61474746001706958876163*10^-6, 7.6439528652715999818366*10^-7, 
 8.674545997849266629631*10^-8, 9.70614819451861140230*10^-9, 
 1.07385078705236470650*10^-9, 1.1771450776261536865*10^-10, 
 1.280485231738244466*10^-11, 1.38386210314581313*10^-12, 
 1.4872688259924495*10^-13, 1.590700145793831*10^-14, 
 1.69415197291341*10^-15, 1.7976210767251*10^-16, 
 1.901104871327*10^-17, 2.00460126236*10^-18, 2.1081085355*10^-19, 
 2.211625274*10^-20, 2.31515030*10^-21, 2.4186826*10^-22, 
 2.522221*10^-23, 2.62577*10^-24, 2.7293*10^-25, 2.833*10^-26, 
 2.94*10^-27, 3.0*10^-28, 3.*10^-29, 0.*10^-30, 0.*10^-30, 0.*10^-30, 
 0.*10^-30, [continuously repeated several times], 
 0.*10^-30, 0.*10^-30, 0.*10^-30, 0.*10^-30, 0.*10^-30, 0.*10^-30, 
 0.*10^-30, 0.*10^-30, 0.*10^-30, 0.*10^-30}

Another is 

Table[N[f[10^x]/f[10^(x + 2)], 30] - 10^2 x/(x + 2), {x, 1, 398}]

Giving:

{5.1843787055343081540041815966, 1.3481548429706546948941035876, \
0.2298032821646331400431563131, 0.0331930367945412883773560502, \
0.0043771580185302427647771435, 0.0005456077208122083756245517, \
0.0000654862190135461596665399, 7.6506284004166399748103*10^-6, 
 8.759504253062014859765*10^-7, 9.87353050091377840035*10^-8, 
 1.09915184581773382945*10^-8, 1.2112619231444879765*10^-9, 
 1.323621030628935636*10^-10, 1.43618247879184902*10^-11, 
 1.5489105631316559*10^-12, 1.661777512922625*10^-13, 
 1.77476140351510*10^-14, 1.8878446949093*10^-15, 
 2.001013187855*10^-16, 2.11425526463*10^-17, 2.2275613279*10^-18, 
 2.340923380*10^-19, 2.45433470*10^-20, 2.5677896*10^-21, 
 2.681283*10^-22, 2.79481*10^-23, 2.9084*10^-24, 3.022*10^-25, 
 3.14*10^-26, 3.2*10^-27, 3.*10^-28, 0.*10^-29, 0.*10^-29, 0.*10^-29, 
 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 
 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 
 0.*10^-29, 0.*10^-29, [continuously repeated several times ], 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 
  0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 
  0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29, 0.*10^-29}

After several more examples and increasing the WorkingPrecession in f, 

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

, I conjectured that,

when x grows without bounds, for integer a,f[10^(x + a)] == f[10^x]/(10^a x/(x + a))

"How can I use it?" You may ask. Lets say you know that f[10^100] is

1.15129254649702284200899572734218210380055074431438648801666395048378\
6304838676240117998602544799162483043795541989292821320520584694737561\
9858787322930686390737864387919272401541317839990869219080546213692318\
3674659470557788570532428107051927066403719152370145446692987566874012\
1236876749481018589885860952810157421023247934838096349920463163787546\
1578221628999375480316143460122987206316610484564925255822975368159940\
8600014247493748226599661796086484246754614415400820331059210126852443\
46028742*10^-98

And you want to find f[10^300]. Simply enter the following:

f[10^100]/(10^200 100/(100 + 200))

giving 

3.45387763949106852602698718202654631140165223294315946404999185145135\
8914516028720353995807634397487449131386625967878463961561754084212685\
9576361968792059172213593163757817204623953519972607657241638641076955\
1023978411673365711597284321155781199211157457110436340078962700622036\
3710630248443055769657582858430472263069743804514289049761389491362638\
4734664886998126440948430380368961618949831453694775767468926104479822\
5800042742481244679798985388259452740263843246202460993177630380557330\
38086227*10^-298

To check it you can use 

In[153]:= f[10^(100 + 200)] - f[10^100]/(10^200 100/(100 + 200))

Out[153]= \
-3.9936179710562538505513847273556460002550958205143231033382996998503\
8874496991966955097579109679315736061442474436408408474886077825412958\
8172768557255993807967123338805984115827197203893069843056570685394386\
5402025212541687556745697311932823550310537084277586254180447711497075\
4099767447229725517466916098168637669296698246841341041914007901211931\
08258666626263607673314627713997808677407174910546*10^-396

Here is one that is harder to check: Find f[10^10^4].:

 f[10^1000]/(10^9000 1000/(1000 + 9000))

giving

:

 1.15129254649702284200899572734218210380055074431438648801666395048378\
    6304838676240117998602544799149170983892021143124316704762732541403378\
    3340995771820280554005251672705721612232540907256079021693931461780377\
    2135564877005554301388608680738152415008390932754134014733922581841389\
    3973253392559061914478264478886900244790291090166024591254691581048354\
    6204893700763843562079932947855669790637611980118379784256624916691170\
    8974106473168612299857252346952803414531680637255002317419426325489273\
    7347972471483650664454679306564260378821462681548700936088037930908316\
    5658184345551002789184065013290800526794356852632582260123160072606109\
    7854185695736227118144576689538022522835215178925366912550322736338922\
    5253866711006115283207141186702073222089255258820260435439407458276917\
    6104809139172406549549568438297438002182819194511022066886420858915917\
    4256821942167283257858214826667344892319761962511086493961589557543089\
    7607403869578190465116556727306663487059584671239762518395549869021179\
    7810281686465709545315559165617055963086909852305386918993024343315452\
    380324954359755288271101245597970868487384568286*10^-9996

You can go very high in the upper end of the partial sums as we do next.

Start with f;

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

Define g such that g[x] is approximately f[10^x]:

g[u_] := f[10^1000]/(10^(u - 1000) 1000/(1000 + u - 1000))

Test it with g[2000], which is close to f

In[198]:= g[2000]

Out[198]= \
2.30258509299404568401799145468436420760110148862877297603332790096757\
2609677352480235997205089598298341967784042286248633409525465082806756\
6662873690987816894829072083255546808437998948262331985283935053089653\
7773262884616336622228769821988674654366747440424327436515504893431493\
9391479619404400222105101714174800368808401264708068556774321622835522\
0114804663715659121373450747856947683463616792101806445070648000277502\
6849167465505868569356734206705811364292245544057589257242082413146956\
8901675894025677631135691929203337658714166023010570308963457207544037\
0847469940168269282808481184289314848524948644871927809676271275775397\
0276686059524967166741834857044225071979650047149510504922147765676369\
3866297697952211071826454973477266242570942932258279850258550978526538\
3207606726317164309505995087807523710333101197857547331541421808427543\
8635917781170543098274823850456480190956102992918243182375253577097505\
3956518769751037497088869218020518933950723853920514463419726528728696\
5110862571492198852630848721557041714332009248522429308500037528856609\
4749743648613906724391357958383583282263392250699192738903186610462300\
0895332393600420259530056414078127754200910666662168913044603358279880\
5862806191309649092141454186597771793091035491858189459159413056753218\
7277091762586467511877225998469693083418839133936608832160695380091439\
2817645297309741677678191880655494400659726751515296070138491930924534\
6565690620482588220344466882352297658647285540827847171287157884738224\
9604520511890163923541465094430697150241483579582055315886422187115331\
9506711825524399063777598716022966937226773473332300851250396513273952\
1135869335466845397627094116507791478689398004144935416040857720399564\
9879199202670714672109123364532456505704001257578092737596024700973977\
9989501767395297612052371846074752388095823469726054788720826089633670\
1770089316794188842635952167515646391606640157327557247401226969263801\
7377661662999356503968839120123291652455084918940959275371845157937758\
4733804404614272539826042782076028823*10^-1997

]

Find f[10^2000] by brute force:

In f[x_] := NSum[(-1)^n (n^(1/n) - 1), {n, x, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 3000]

In[202]:= f[10^2000]

Out[202]= \
2.30258509299404568401799145468436420760110148862877297603332790096757\
2609677352480235997205089598298341967784042286248633409525465082806756\
6662873690987816894829072083255546808437998948262331985283935053089653\
7773262884616336622228769821988674654366747440424327436515504893431493\
9391479619404400222105101714174800368808401264708068556774321622835522\
0114804663715659121373450747856947683463616792101806445070648000277502\
6849167465505868569356734206705811364292245544057589257242082413146956\
8901675894025677631135691929203337658714166023010570308963457207544037\
0847469940168269282808481184289314848524948644871927809676271275775397\
0276686059524967166741834857044225071979650047149510504922147765676369\
3866297697952211071826454973477266242570942932258279850258550978526538\
3207606726317164309505995087807523710333101197857547331541421808427543\
8635917781170543098274823850456480190956102992918243182375253577097505\
3956518769751037497088869218020518933950723853920514463419726528728696\
5110862571492198849978748873771345686209167058498078280597511938544450\
6597762783139495433741884790474781454862846244198223401237480573674236\
4463118953370361234107504746499839506070180488466274310861497007708124\
7531049171254247021317442285517029952373158629538157258925201464308883\
7261378646935237069028384362414719458837579109071459566582713060290759\
7715020477289934522840601988588807812532803919543964899917097014996009\
1523772523122916963890276166953764687454481935787728341490205835353563\
7037027325006181019939423744971861820938280603244268309275367385585320\
0352025010346363364340939265951674785944654923153407950025462218746638\
8987981096603956474219740328144754031639588061034476483971049028388089\
5429349924672282498515907802777007046001226977569476462738627441597792\
0996129309643609179395618950988275761390703474038532538418642758726920\
0745287855627273080001366829321525609421177861900201909932368162922731\
6134550672707339313262595430967497468392447545239327700742210136269769\
9448315164406666838140953010567070530204160252020410521167407092133153\
9919976753637811218655111640963566617960064955222454518256143646556562\
3319279886971083752500630577450217787532945319539286266762458212631657\
8956562615955112869175663341858392950377027028587440156175937426523967\
5992486669665535908081093963673475788626352572086040171478034965548633\
3275502065293770553964589862333894362557313336100252034738702306866723\
7252231900771628279428575631116592820122177616389957126689432290690530\
2126577063928469621518520649725420093449670517852034635099462749636548\
2290064716060616844360660926298562988720899644831700021688560753298998\
9209982913635406738382792962581607465767405435400124459385726984354481\
2846518246662963758480026314121617338215950137737517163117051770148727\
9850770249214664296052762392324100847279871129429253715780855091816115\
2045192252316069181660245026216345222368650054851109042984141331103190\
7942638261038253291471354448201102654558086418437345524590936214590694\
934638733489475106396761915692962301680633663763905699880*10^-1997

They are roughly the same to 1000 significant digits:

In[207]:= Out[198] - Out[202]

Out[207]= \
2.65209984778569602812284219002435102790252559031215881519808654744112\
9064947316790880182740054600650096933766570603678806364322134402300590\
2542255166757828824813073017819589460218310635057175583317570200554020\
7082401190108074184071787686232003220023421159244433500157131156512304\
4284884163605497362458126002486514926557798231980067951026248200198071\
5483758989206668658812692283197133117022139491592852550419180973596712\
5645419071539853297119280360504011882979695204938466125674931868839829\
0360204134945883532930320297633778700661105480153001191546868151780356\
9943665945007129215128211855017889290122493429452731321478882388628889\
2340735378836303744704980994311045893206980869201147544498492779984321\
7359321556175544945970277428000861627485739725937618589933724577516884\
3265675289508647662670511999568752225030218333090675010248014611669157\
6263458533819412078218546229542735533746885880634107012431109902920171\
9070624368915579418406263737370163157462963502166798852854892402076057\
01685089771508958293*10^-2994

You can even find g[10^6]:

In[208]:= g[100000]

Out[208]= \
1.15129254649702284200899572734218210380055074431438648801666395048378\
6304838676240117998602544799149170983892021143124316704762732541403378\
3331436845493908447414536041627773404218999474131165992641967526544826\
8886631442308168311114384910994337327183373720212163718257752446715746\
9695739809702200111052550857087400184404200632354034278387160811417761\
0057402331857829560686725373928473841731808396050903222535324000138751\
3424583732752934284678367103352905682146122772028794628621041206573478\
4450837947012838815567845964601668829357083011505285154481728603772018\
5423734970084134641404240592144657424262474322435963904838135637887698\
5138343029762483583370917428522112535989825023574755252461073882838184\
6933148848976105535913227486738633121285471466129139925129275489263269\
1603803363158582154752997543903761855166550598928773665770710904213771\
9317958890585271549137411925228240095478051496459121591187626788548752\
6978259384875518748544434609010259466975361926960257231709863264364348\
2555431285746099426315424360778520857166004624261214654250018764427584\
6784740139800859592078235584135646781413130043752666984334071191974640\
8597345629156414605725368196863381499986518832614544218899402121570851\
0583484314825156735451762546018264566784175380012768153949683431808627\
6707784179416647362077298983902270631627458619607454614331281338063453\
9745072146070727377342847166021803510111348189465286859811382754396221\
9810872470659241419740542185617689533943526740705434689987255226074533\
5052710374422824756001554307391516627242372042424158095047514010459076\
1118612697619415666900335253575892598450897068916224663119038275771039\
8822208932198914366064045120549263224186388562794230201147657514349147\
8702047116746009949603372041786877463805128579260619225159218893629348\
1669369211025180448898759137488067056314556973466484397671053578903747\
2816989590802198065726502480257746798928814706595635081058179251045401\
9741204083242715668792857336887486145616201703384403903671643414224178\
1374921765339928869471569450206142876196339603638185498109247368585850\
1102137643872877779795185542148189589508317900410196654075841920570505\
8035783932164047241489886220068322075466433817129459099327848898516712\
2990652943154782979605947777851836455553306652837625492244864137518162\
8254731203738671524937211214048507686909327316872709668078996906356907\
0084386824015444873319043112169071770894839562034553216051946446991161\
8982154884066067649101660172616896157938362223520016131498434680607794\
7168469154444370894904516106355520899732120597846314907270633204412067\
0551262582600076702135952973000844301325861102671945362519066907360149\
1854719018651219245184703045374222235310826678741133619750748757108910\
2838603865119913021777099938973412343930576337258793240730758180417969\
0631836387103952358870387060819951415410587067959835630943871415130094\
6755374937094870283301344964050976394345960293060753909215251210518251\
7721704409528511110551542235976353151044306919309775008418995463569530\
771521779231523353086127438782428006385677607191180850829*10^-99995

[Edit Any suggestions here about g? I actually wonder about it!]

{Edit: I removed some rambling.]

TO BE CONTINUED! and possibly edited.
POSTED BY: Marvin Ray Burns
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I think I found something interesting!

First lets define f to give several digits from the upper end of the partial sums of the MRB constant :

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

Define g such that g[u] is approximately f[10^x]:

g[u_] := (f[10^1000]/10^(-997 + u))*u:

Discover that f[10^10^x]*10^10^x is isomorphic to 10^x Log[10]/2, as x gets large. I mean the mantissa of f[10^10^x] becomes Log[10]/2, as x gets large:

In[69]:= N[Log[10]/2, 20]

Out[69]= 1.1512925464970228420

In[57]:= Table[N[g[10^x], 20], {x, 0, 5}]

Out[57]= {0.11512925464970228420, 1.1512925464970228420*10^-9, 
 1.1512925464970228420*10^-98, 1.1512925464970228420*10^-997, 
 1.1512925464970228420*10^-9996, 1.1512925464970228420*10^-99995}

Discover when f[10^10^x]*10^10^x == 10^x Log[10]/2:

In[62]:= Table[N[f[10^10^x]*10^10^x - 10^x Log[10]/2, 20], {x, 1, 3}]

Out[62]= {1.3805391560885809316*10^-8, 1.3312059903520846169*10^-96, 
 1.3260499238928480141*10^-994}

Try it for f of great value:

In[60]:= Table[N[g[10^x]*10^10^x - 10^x Log[10]/2, 20], {x, 0, 8}]

Out[60]= {1.3260499238928480141*10^-997, 
 1.3260499238928480141*10^-996, 1.3260499238928480141*10^-995, 
 1.3260499238928480141*10^-994, 1.3260499238928480141*10^-993, 
 1.3260499238928480141*10^-992, 1.3260499238928480141*10^-991, 
 1.3260499238928480141*10^-990, 1.3260499238928480141*10^-989}

I think the last set of answers is governed to "limit out" in the neighborhood of 10^-996 by the formula for g. Any help here would be great!
POSTED BY: Marvin Ray Burns

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!
POSTED BY: Marvin Ray Burns

f[x_] := NSum[(-1)^n (n^(1/n) - 1), {n, x, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 400]

As x grows without bound f[10^x] == 10^-x x Log[10]/2 leads to the following acceleration method for the partial sums of the MRB constant

enter image description here

Here is an example:

In[225]:= 
m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}, 
   Method -> "AlternatingSigns", WorkingPrecision -> 400];

standard[x_] := (NSum[(-1)^n (n^(1/n) - 1), {n, 1, 10^x}, 
   Method -> "AlternatingSigns", WorkingPrecision -> 500])

plus[x_] := (standard[x] - 10^-x x Log[10]/2)

Table[N[{(m - standard[x]), (m - plus[x])}, 30], {x, 0, 100, 10}]

Out[228]= {{0.187859642462067120248517934054, 
  0.187859642462067120248517934054}, \
{-1.15129254776743274319429239444*10^-9, \
-1.27040990118529666709714903244*10^-18}, \
{-2.30258509299404568453691834027*10^-19, \
-5.18926885582869572712277056422*10^-38}, \
{-3.45387763949106852602698718214*10^-29, \
-1.17590768666018420974624000155*10^-57}, \
{-4.60517018598809136803598290937*10^-39, \
-2.09798339326141874738448664089*10^-77}, \
{-5.75646273248511421004497863671*10^-49, \
-3.28515400538657318555081659969*10^-97}, \
{-6.90775527898213705205397436405*10^-59, \
-4.73741952303564752424522987791*10^-117}, \
{-8.05904782547915989406297009140*10^-69, \
-6.45477994620864176346772647557*10^-137}, \
{-9.21034037197618273607196581874*10^-79, \
-8.43723527490555590321830639266*10^-157}, \
{-1.03616329184732055780809615461*10^-88, \
-1.06847855091263899434969696292*10^-176}, \
{-1.15129254649702284200899572734*10^-98, \
-1.31974306488711438843037161851*10^-196}}

That table looks like

enter image description here

We see,,for x>0, the simple 10^-x x Log[10]/2 increases the accuracy exponentialy!!
POSTED BY: Marvin Ray Burns

Here is a twist with an interesting property, where previously enter image description here

Now let

enter image description here

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:
POSTED BY: Marvin Ray Burns
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