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.