# MRB constant approximations using TranscendentalRecognize

GROUPS:

## By using TranscendentalRecognize, I think I came up with the most efficient approximation of MRB from given forms.

It might help in analysing the code's efficiency by checking out he attached sample MRB inver...nb .

I started with a program posted online by someone going by the name of Simon. . A similar one is found deep within in Mathworld.and at http://forums.wolfram.com/mathgroup/archive/2005/Jan/msg00254.html .

 TranscendentalRecognize[num_?NumericQ, basis_?VectorQ] :=
Module[{lr, ans},
lr = FindIntegerNullVector[Prepend[N[basis, Precision[num]], num]];
ans = Rest[lr].basis/First[lr];
Sign[N[ans]] Sign[num] ans]


I built around it a data mining operation.like the following, where h is a list of many constants of the form x(1/x). I lost the list for h there, but it had 40 or more constants, like, h = Table[x^(1/x), {x, 2, 40}].

m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity},Method -> "AlternatingSigns", WorkingPrecision -> 1000]; .

.

      d = 100; er = 20; Table[{Print[x, y], (t =
TranscendentalRecognize[N[(m), d], c[x, y] = h[[x ;; y]]];
s = N[m - t, d]); If[s^2 < 10^-(2 d + er), Print[{t, s}]]}, {x,   1,  10}, {y, 10, 10}]


I got some, what I think are, efficient ( i.e.fewer variables with smaller constants), approximations to my MRB constant. (MRB constant is defined in Wolfram' MathWorld and is known by W[A..) These all were limited by only using the form of c^(n/x), where n and x, both<100.

In[197]:= Quiet[
N[m - (-129858773922357615372945307143544254 -
332618118135196201861563173048187520*3^(1/5) +
360989141074787715535168098417020609*2^(2/5)*3^(1/5))/
261711912538111957032871243762602971, 200]]

Out[197]= \
3.67832800603717470665368223835575291198522877559637247418956870295862\
1540717957587861917095449501665482864609259912010997464854189390496601\
7326568194578935410565495353817533097199630888364091944358909*10^-144

In[197]:= Quiet[
N[m - (-129858773922357615372945307143544254 -
332618118135196201861563173048187520*3^(1/5) +
360989141074787715535168098417020609*2^(2/5)*3^(1/5))/
261711912538111957032871243762602971, 200]]

Out[197]= \
3.67832800603717470665368223835575291198522877559637247418956870295862\
1540717957587861917095449501665482864609259912010997464854189390496601\
7326568194578935410565495353817533097199630888364091944358909*10^-144


Here are a few more approximations that I found, which ended up giing a lot of digits more that the value I gave to my mining program: (These are limmited to a few other restrictions.)

   In[218]:= Quiet[
N[m - (791403284963475530487868944442284292407553*StieltjesGamma[2] +

3364575415987830296212016870286414208213855*StieltjesGamma[4] +

6501952717766008515310532964426708542883509*
Sum[8^(-n!), {n, 1, Infinity}] +

2814879473510959962780731825640664542716703*
Sum[9^(-n!), {n, 1, Infinity}])/
6717984055252083042876120552656613060137774, 300]]

Out[218]= \
-6.2242642909663704162355532151579039524967358848368194746368886689290\
751661818839021804\
93643940426236207972707862693901673385279106988471319969698985140112\
7435709703873332134\
.9155634653911*^-216

\

In[219]:= Quiet[
N[m - (-372967063067619742205241026585337180*StieltjesGamma[2] +
320687606599321465073123905986839263*StieltjesGamma[4] +
102211052827902295002288334566721956*StieltjesGamma[12] +

212441894526555216749911459715341854*
Sum[8^(-n!), {n, 1, Infinity}] +

155185652908810652615766086341025368*
Sum[9^(-n!), {n, 1, Infinity}])/
284315814249073009815326355615579013, 300]]

Out[219]= \
-9.9524676215315096691725832693825358984906951576789134259705676021664\
248059747678548554\
60610615180989951232428253214135672396501134144042507059888478602960\
236138224312425134.\
12203729059152*^-217

Null

In[225]:= Quiet[
N[m - (7104443136721728575905321650535196463313633*3^(2/9) -

2180718111811910815323291747705043372745259*StieltjesGamma[9] -

2896588673082600778149973623311628930807255*
Sum[12^(-n!), {n, 1, Infinity}] -
6015801821759926483557786721397041400269244*Zeta[3])/
8390253020596747014154150758555797074475505, 300]]

Out[225]= \
-3.3222593389038579058691628775383126316686843433100927697761938470770\
423450105378910571\
33932112092296214244839309765166771670398490360706592080820770773400\
60779447336714132.8\
8558801750568*^-217


Here is another one , I think is good, Those of you that might know how to judge such appox. ,how am I doing? Because

TranscendentalRecognize[],, I think, was made to give the optimized approximation I will attach a few of my number mining notebooks.with the most precise approx.the precision in10^-x will be named that x.

Attachments:
2 years ago
4 Replies
 I've found too many interesting forms for many digits of the MRB constant.I looked at a few, just as good, for other constants too.Here is one that is extra special to me-- I don't know -- some would just see that I added a bunch of random numbers and got an answer..I did a little studying about the TranscendentalRecognize function. With the restrictions given it was supposed to give the "simplest" solution for the given number digits. z = {\[Pi], E, GoldenRatio, Khinchin, EulerGamma, GoldenRatio, Catalan, \ Sqrt[2], 3^(1/3), Sqrt[2], 5^(1/5), 6^(1/6), 7^(1/7)} TranscendentalRecognize[num_?NumericQ, basis_?VectorQ] := Module[{lr, ans}, lr = FindIntegerNullVector[Prepend[N[basis, Precision[num]], num]]; ans = Rest[lr].basis/First[lr]; Sign[N[ans]] Sign[num] ans] For[y = 3, y < 11, Print[y]; p = TranscendentalRecognize[N[m, 20 y], RandomChoice[z, 4]*y]; Print[p]; Print[e = N[p - m, 64]]; y++] Which factored by my observation is Which equals, 9.67609665331181132 ... 05184839719402143746300.*^-201 The result::: 10539265076340778455746831348074621/1147054894409218164005683918321030 +(1315939211433987901830906252006812 Sqrt[2]) / 114705489440921816400568391832103 +(2238154781652703157536445225769983 Sqrt[5]) / 229410978881843632801136783664206 +(464968409472439567819541453580715 6^(1/6)) / 114705489440921816400568391832103 -(1917024616230569159116802658552184 Pi ) / 114705489440921816400568391832103 I do realize, however, even if this is the simplest approximation,to that many digits, of that particular form, there is no chance of being the simplest form to use! P.S.I got a little exotic and found a nice closed form approximation to m^m^m^...... mr=(3266780045684960162847550288713990225631168+ 1110218827497735378447017478768998194987712 Sqrt[2]+ 5397133631025017706929697830832340263986880 3^(1/3)- 6209268135253131845504446727054527890012160 5^(1/5)- 3722403313326121192182250135829290878917440 6^(1/6)+ 10028893256036615933248156928358431064871936 7^(1/7)- 4511138177384336628384814819384709016906304 E)/ 35171415528443234478484447899288103133690 To check it I will use a tetration law, I don't know the name, But you can find it in MathWorld on the tetration page.  N[m - mr^(1/mr), 400] Out[65]= 1.\ 8227207233303348653454461326515515800093235125329919778950382492329113\ 4594869822885566472851208477062791211814387757010664442953073549308007\ 6927632430241699590693763805649694595954601441174700087168936922446384\ 3118477884367149009785953875366770689466302668084408526394897438402179\ 4398904288853785020951207815180537234335182744342204474468543109558551\ 6156647729629573414184056008550101815194046180953*10^-325 By the following input I GOT A NEARLY 1,000 DIGIT APPROX In[74]:= z = Join[{I, \[Pi], E, Sqrt[2]}, Table[x^(1/x), {x, 1, 15}]] Out[74]= {I, \[Pi], E, Sqrt[2], 1, Sqrt[2], 3^(1/3), Sqrt[2], 5^( 1/5), 6^(1/6), 7^(1/7), 2^(3/8), 3^(2/9), 10^(1/10), 11^(1/11), 2^(1/6) 3^(1/12), 13^(1/13), 14^(1/14), 15^(1/15)} TranscendentalRecognize[n_, basis_] := Module[{c, d, digs, e, id, lat, powerten, r, s, vals}, {d, e} = RealDigits[n]; s = Sign[n]; c = FromDigits[d]; powerten = 10^(Length[d] - e); digs = (RealDigits[N[#1, -e + Length[d] + 5]] &) /@ basis; r = (FromDigits[Take[First[#1], -e + Last[#1] + Length[d]]] &) /@ digs; Tlat = Transpose[ Append[IdentityMatrix[Length[basis] + 2], Flatten[{powerten, r, c}]]]; vals = Take[First[LatticeReduce[lat]], Length[basis] + 2]; Expand[-((s (Take[vals, {2, -2}].basis + First[vals]))/Last[vals])]] In[75]:= For[y = 110, y < 2200, Print[y]; p = TranscendentalRecognize[N[m, 5 y], RandomChoice[z, 7] y]; Print[p]; Print[e = N[p - m, 64]]; y++]; m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 1000]; In[87]:= Block[{ \$MaxExtraPrecision = 2000}, N[m - (732342421548474675180526599895409684264644327151934824164674474461\ 7922491658539305019493955220461660037676829012116678161973717 + 4072149642232149576053764120676969872061896542320267666183500991\ 02049588747083043017470783162958681270629759687955222181909461330*2^(\ 3/8) + 7681034247847453546335061180942226612630628658675512251591975879\ 27899297132477996528019803100765350880729688992218863454548508878*2^(\ 1/6)* 3^(1/12) - 34045134946025441841011260331526924008281141115692566262609055\ 8152409346539144337134261814844546786394542003505888957271010500854*3^\ (2/9) - 609926526794066808991516705993794777295179002582017985851779358\ 782777912145100024260322124118925720144824696881973453747630696704*7^(\ 1/7) + 11372239599441213960858419663818160615356886708217344335954638\ 9846875907225388509126032751322453242940643260453219863039395130962*\ 13^(1/13) - 139334155741910768876027728909033188772221814812961177310973301\ 700033416896043657269919708071396437672738505935341191663898499044*E)/ 196686824825458435614269629594068012795109891362859180654527231644\ 718522850534710213275151197132395131009154363507115493359209, 1000]] Out[87]= 3.\ 0284237122944438589618739933075298246671309372966698310863473780943006\ 37126254014208777456974745659*10^-997
 Here are my latest approximations to the MRB constant.They are the best looking one out of a few thousand. These approximate 1000 digits of the constant, but give the most beyond, 1000 digits, accuracy. N[m - (510017535631180687460856957450572305326540096915661936978654660685564647512350782555553614907470983387825537836511551063206613482712467827893\ 01468559609485014131037773093095425665777333045322543428553522*(-1 + Sqrt[2]) - 96068150340234077374726370910347925333938216211589364597933496589097\ 075853239867010973807246924811862210135463028988657144609634882312183141641523864425497419159770197204815060852100498555727184938689116* (-1 + 2^(1/18)*3^(1/9)) + 23333703534904618706173261841895326308700547835506225964185595796344451478775469597207817863820838379134974452053729\ 3369333105298477043109281116710787527167404406291408274367957775851669383906747626354830*Sum[4^(-n!), {n, 2, Infinity}] + 40313385556940898081252974448093819151002029893520177263320252381062732980131887242120530425911918252062181272894937220571111439759199788063701\ 4439799379786658216842616733570082844959130397748389631088288*Sum[14^(-n!), {n, 2, Infinity}])/11226020246276929216469224351763847870811895611134682\ 9809426965838194867887854374179420796093728124712438534958688705043058060873338334003999743465958730946783909428577286552639810756878788810016950541\ 959, 2000] . It gives9.1978955072905472962...2*10^-1018 . N[m - (404355126688171915451034416312230604919707970452383451853925931\ 40406938951988728889681015103623256198012637149747345520129713\ 49689454162107757602250020811957699990816546*(1 - 7^(1/7)) + 33051480591839495225039933253849294611927448814171710704107740940\ 734665\ 040457829733431797688126677213799052071795406260439448132384992\ 74294322897183697379311258530999832*(-1 + 14^(1/14)) - 24189499316961197596532386979514491525599470961143368177569250102\ 82955146179847110763589857004031035173663050083808012723899090\ 473565355251693036218605461019069457446148*(1 - 17^(1/17)) + 82468081425975915242688262123067077311822879410444962143373037763\ 622634\ 606906917316082600654001559505105511211964587221098020620728591\ 82678249467359584256403093648400278*Sum[2^(-n!), {n, 2, Infinity}] - 23196615765931373730391709909338745951421286383132885907911476173\ 69723860751338004736792098716384442993092400628717132353654831\ 279801942857974608507685271164128259600611* Sum[20^(-n!), {n, 2, Infinity}])/ 10716692834803667275169650412872773031712463947868978911\ 401491029190218128259539722750075189375635799016814648578340611058\ 418703882776034702173059087881414809009655698842, 2000] gives 1.29057693406447674817351694260292503364903562408778360761951285430083\ ... 562007060546444086256424992932423084037209691675528920*10^-1017 and N[m - (479726275966738930950716668460453456135609546153820124961462648\ 4512655106535889637667232127769509716461740872574122078642387776516851\ 09388282\ 12443377294736414176483302924133740398634813902217928087638057* Sum[4^(-n!), {n, 2, Infinity}] + 5838998640136470517323044439381905625083886232218587\ 669749531087797743382913252545417668577092986554098832543936290\ 9113467768066621051147331513413336416581833310908902378720986774747056\ 064016042598151\ 125*Sum[7^(-n!), {n, 2, Infinity}] - 63218949949966052258049515603589770732070427320021246973471652245\ 5144788248490957660650633781863001770791932590\ 071033829885733702925886237886192728969625214564447134735540638\ 5925595638579007900489820209*Sum[15^(-n!), {n, 2, Infinity}] + 15198929012735579673282610542007594760375937317388650950231682497\ 1899195175150323553974961146574023964586942011016795308214767786746900\ 16895505\ 392739086196475640858326345017228496199996878438805028527771* Sum[17^(-n!), {n, 2, Infinity}])/ 224988407960821873723472823677123883665729840705233851\ 269518866426796111745109675950172197114835253338797358527687132274\ 3557882097017477522481690748025398465304015951108345956023807857443692\ 657239661543\ 2, 2000] gives 2.57879929491783173010969833912581455697003587799754559692884173462338\ --- 578974308935437637235778544541222839048731919691392341242288831*10^-\ 1017 Also N[m - (676553166992866963815372906222646303607573324645363021255484344\ 5847651711736285938683235621024058639551951835735298345223727168642453\ 37735108\ 73613949894745859102893275100501442382626915575445148788736214*\ (1 - 11^(1/11)) - 14619701004192034806300676602885461655161865631216564768365650294\ 65\ 076216299072104991168738127984191377243798233621863800457100013\ 0509345301162175372820032729558482082979094524738063563853873225007240\ 4928* (1 - 19^(1/19)) + 68539239911237007645329097139664653912687801952352131936740047785\ 40357555041564396019643409050283620285059876883679335919953\ 977606127636155870844440362109900051767228224136857904246430358\ 8400719638395372*Sum[13^(-n!), {n, 2, Infinity}] + 51631709973922849174083847516931964611996582831207246415493453517\ 1360384966066875043647023902955518731996597302090507331070815563491676\ 50911951\ 564015771281862255807366068747634361550836780876919887207074* Sum[16^(-n!), {n, 2, Infinity}])/ 459606656778282203777615873161355445177495913644534440\ 236428009020039867679836025106859862935288356868529917262162124684\ 8655339707929850088398876890644711371164045251650873684631194412373318\ 788116834130\ 5, 2000] gives -3.6102997014559870502180584574338386601100194544114033623209409990651\ .... 4013259366951605041002710996897846676570391136242678653046064605*10^-\ 1017 N[m - (-14059768861495021278710761814432661641419385394995410885384053\ 4321718939400795528466474229654128294482783893218374581982359790435922\ 70347287\ 850912019400773283925905864775*(1 - 13^(1/13)) + 74819870702135710155830108496241896819204806952794126321695372878\ 4421980599403317506355275210308751\ 837169775662594125358889924021938820605278128626037787819772197\ 2689543*(-1 + 14^(1/14)) + 1026745578533744923810544980109604285225991455455168848218\ 851940485590090158684730723388336061515548637248082857303503426\ 8011245942869978406865653859126448947036699880250* Sum[5^(-n!), {n, 2, Infinity}] - 23919308882782978080426273494605406622652109218599068941380171181\ 0873283701722728640061108639572805924891231810861486595861053426099988\ 74913830\ 397311908108691461538843588*Sum[11^(-n!), {n, 2, Infinity}] + 33490640981554250803555955384317846792767847299010359091716296701\ 777133181948335535665\ 363570732763918461558209511660291118481482138653844052627227766\ 575260126610639969338*Sum[16^(-n!), {n, 2, Infinity}])/ 264197139971924290274318703857\ 045641213393263172333862948269611997449923895383664022896461113170\ 0352455914156203328519044605284050440986586394175399184484438567395205\ 6001, 2000] gives 2.95037252497728220301535655036996123361794994342988068078317201753568\ 5315152510783969290388149141412681616174537035046372757542091836244471\ 0147825110223321183122124086148527098121970008184357909548197*10^-\ 1017} 
 Below and attached are two huge approximations of the MRB constant using only integers and the form Sum[x^(-n!), {n, 2, Infinity}],, for x from 2 to 101..This first one gives around 3000 accurate decimals.And the one below it gives around 4000 accurate decimals! In[7]:= TranscendentalRecognize[num_?NumericQ, basis_?VectorQ] := Module[{lr, ans}, lr = FindIntegerNullVector[Prepend[N[basis, Precision[num]], num]]; ans = Rest[lr].basis/First[lr]; Sign[N[ans]] Sign[num] ans] m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 4000]; s = Table[Sum[x^-(n!), {n, 2, Infinity}], {x, 2, 102}] In[50]:= TranscendentalRecognize[N[m, 3000], s[[2 ;; 101]]] Null N[m - (-1617601210758780475076147232954091* Sum[3^(-n!), {n, 2, Infinity}] + 1260730972958628376379103266584756* Sum[4^(-n!), {n, 2, Infinity}] + 168255942381333226880232447282467* Sum[5^(-n!), {n, 2, Infinity}] + 526758508745229051826843765249190* Sum[6^(-n!), {n, 2, Infinity}] - 2670042164293488356909920338891358* Sum[7^(-n!), {n, 2, Infinity}] + 3829272605182291498586693806942722* Sum[8^(-n!), {n, 2, Infinity}] + 859415018288984049908165889504129* Sum[9^(-n!), {n, 2, Infinity}] + 3980672361148310552936792479918956* Sum[10^(-n!), {n, 2, Infinity}] + 2949125963814399232221423020582609* Sum[11^(-n!), {n, 2, Infinity}] - 1704497602955159964193025017671421* Sum[12^(-n!), {n, 2, Infinity}] - 1158615646978266430347663044541968* Sum[13^(-n!), {n, 2, Infinity}] + 2940566178482760393686097895245937* Sum[14^(-n!), {n, 2, Infinity}] + 3537625077232750001025112741131143* Sum[15^(-n!), {n, 2, Infinity}] + 1628630162143702095790590207317237* Sum[16^(-n!), {n, 2, Infinity}] + 2075441527976471396863264573730348* Sum[17^(-n!), {n, 2, Infinity}] - 2264173752934939591163612937875735* Sum[18^(-n!), {n, 2, Infinity}] - 2522296388129947914734380874599382* Sum[19^(-n!), {n, 2, Infinity}] + 2320118429832380290630535582086539* Sum[20^(-n!), {n, 2, Infinity}] - 2463815400152015299479476128150570* Sum[21^(-n!), {n, 2, Infinity}] + 2638986590182329976236035098810060* Sum[22^(-n!), {n, 2, Infinity}] + 3251654159711357468899799174679152* Sum[23^(-n!), {n, 2, Infinity}] + 93915231319017548602476626871287* Sum[24^(-n!), {n, 2, Infinity}] + 193808727512029207895977355455647* Sum[25^(-n!), {n, 2, Infinity}] + 3607117201144790057874151852014759* Sum[26^(-n!), {n, 2, Infinity}] + 3262007418561507649847272272032955* Sum[27^(-n!), {n, 2, Infinity}] + 2480402689383408099208714687051739* Sum[28^(-n!), {n, 2, Infinity}] + 1308509867990398861324717427268301* Sum[29^(-n!), {n, 2, Infinity}] + 1311639103727367378793170201050131* Sum[30^(-n!), {n, 2, Infinity}] - 1648727214133032285519999314329705* Sum[31^(-n!), {n, 2, Infinity}] + 3895395322625799512405275036584486* Sum[32^(-n!), {n, 2, Infinity}] - 4228124835647564954643945470573976* Sum[33^(-n!), {n, 2, Infinity}] + 1724355876667841445871244995563479* Sum[34^(-n!), {n, 2, Infinity}] - 821120993451512960591013772378796* Sum[35^(-n!), {n, 2, Infinity}] - 2219010432574623362013336951784027* Sum[36^(-n!), {n, 2, Infinity}] + 955107235645995663705816457381081* Sum[37^(-n!), {n, 2, Infinity}] + 4977626512233618799592076868098467* Sum[38^(-n!), {n, 2, Infinity}] + 479750350430292719426344888964982* Sum[39^(-n!), {n, 2, Infinity}] - 1009428065023075859234682582260676* Sum[40^(-n!), {n, 2, Infinity}] - 453768351398454243808381569101699* Sum[41^(-n!), {n, 2, Infinity}] - 804331952205012340117978795714571* Sum[42^(-n!), {n, 2, Infinity}] + 2112537773325380956038299597223034* Sum[43^(-n!), {n, 2, Infinity}] + 1563909600882194431077161802113231* Sum[44^(-n!), {n, 2, Infinity}] + 3281512201875089338818140791672361* Sum[45^(-n!), {n, 2, Infinity}] - 841693145563449440980152624889040* Sum[46^(-n!), {n, 2, Infinity}] - 1180596415104015890930529257781618* Sum[47^(-n!), {n, 2, Infinity}] + 256894002482080676102804595099554* Sum[48^(-n!), {n, 2, Infinity}] + 3059743069488164456875317821197684* Sum[49^(-n!), {n, 2, Infinity}] + 618378447020184993694643494855854* Sum[50^(-n!), {n, 2, Infinity}] + 1207849584860413404886567472602012* Sum[51^(-n!), {n, 2, Infinity}] + 3269594033521468442241723242546780* Sum[52^(-n!), {n, 2, Infinity}] - 519110770734317313929892528340481* Sum[53^(-n!), {n, 2, Infinity}] + 5337933979867080683965490693782692* Sum[54^(-n!), {n, 2, Infinity}] - 845305587078512215850327456666298* Sum[55^(-n!), {n, 2, Infinity}] + 3638411700443380962703568627301563* Sum[56^(-n!), {n, 2, Infinity}] + 57104890901297559220617943398070* Sum[57^(-n!), {n, 2, Infinity}] - 349193535926582284655062345576058* Sum[58^(-n!), {n, 2, Infinity}] - 2574102460519834845887788602861507* Sum[59^(-n!), {n, 2, Infinity}] + 626918354620070457316744226035168* Sum[60^(-n!), {n, 2, Infinity}] + 925910782814132818900483793749726* Sum[61^(-n!), {n, 2, Infinity}] + 2659043142343524075790297170462132* Sum[62^(-n!), {n, 2, Infinity}] - 3776624195115615476424023462853132* Sum[63^(-n!), {n, 2, Infinity}] + 289252819950910792633107902104858* Sum[64^(-n!), {n, 2, Infinity}] - 4335832676811110969381326369850434* Sum[65^(-n!), {n, 2, Infinity}] - 1049666000572944258193455109018929* Sum[66^(-n!), {n, 2, Infinity}] - 2060181142655155567883399336847960* Sum[67^(-n!), {n, 2, Infinity}] - 1802686787596758226171368511609941* Sum[68^(-n!), {n, 2, Infinity}] + 560149266456680122048780702178403* Sum[69^(-n!), {n, 2, Infinity}] + 45326892699113686458133865345872* Sum[70^(-n!), {n, 2, Infinity}] - 2189834207054837650558549556270607* Sum[71^(-n!), {n, 2, Infinity}] - 200007960908326787314318800286238* Sum[72^(-n!), {n, 2, Infinity}] + 1906908709896907749806302689843055* Sum[73^(-n!), {n, 2, Infinity}] - 2413831603161356405527598276237701* Sum[74^(-n!), {n, 2, Infinity}] + 349851432963186321571769787692286* Sum[75^(-n!), {n, 2, Infinity}] - 1409077148223763233830923073076654* Sum[76^(-n!), {n, 2, Infinity}] - 3258926296143830773420165267895655* Sum[77^(-n!), {n, 2, Infinity}] - 2788017156113146729276123053829960* Sum[78^(-n!), {n, 2, Infinity}] + 2692847787140868892915206384398765* Sum[79^(-n!), {n, 2, Infinity}] - 567067016846573451010485778130831* Sum[80^(-n!), {n, 2, Infinity}] - 1905831513124390141890451640600611* Sum[81^(-n!), {n, 2, Infinity}] - 5136597947331669055009371904973487* Sum[82^(-n!), {n, 2, Infinity}] + 912278368494920565761271769690129* Sum[83^(-n!), {n, 2, Infinity}] - 2580979677015581716819042733783251* Sum[84^(-n!), {n, 2, Infinity}] + 1884158460408959578540033438166537* Sum[85^(-n!), {n, 2, Infinity}] + 2996809848467580062948210521300363* Sum[86^(-n!), {n, 2, Infinity}] + 2547118062655141771555127378895892* Sum[87^(-n!), {n, 2, Infinity}] + 122733378092474394591379091380222* Sum[88^(-n!), {n, 2, Infinity}] + 1609043837874589833719447903600838* Sum[89^(-n!), {n, 2, Infinity}] + 706449525697658734698912642263703* Sum[90^(-n!), {n, 2, Infinity}] + 331198650888790532168855884736297* Sum[91^(-n!), {n, 2, Infinity}] + 616901082829008843727297647565846* Sum[92^(-n!), {n, 2, Infinity}] + 2407854441179185932069643380268222* Sum[93^(-n!), {n, 2, Infinity}] - 4494561604681894784950422947849268* Sum[94^(-n!), {n, 2, Infinity}] + 1259577304594505752616397860280623* Sum[95^(-n!), {n, 2, Infinity}] - 2564935607306630357867809604407419* Sum[96^(-n!), {n, 2, Infinity}])/ 230024844450836751070309032939613, 3000] giving  Out[49]= 1 3602520033281506638072983352187247657142056301838474533913712593885236357268243847.\ 36565286514424*^-3002 Here is the 4,000 approximation: N[m - (-296516481636968137503982279733117020027871* Sum[3^(-n!), {n, 2, Infinity}] + 325661615276655660175959176674077806917110* Sum[4^(-n!), {n, 2, Infinity}] + 642182094925939713175056356584465598954027* Sum[5^(-n!), {n, 2, Infinity}] + 601175274312643299513368500777196611943043* Sum[6^(-n!), {n, 2, Infinity}] + 461071564223118140831888034215659217102634* Sum[7^(-n!), {n, 2, Infinity}] - 46056184702533685913819676113042970084596* Sum[8^(-n!), {n, 2, Infinity}] + 279339004802534787952844028108115284244029* Sum[9^(-n!), {n, 2, Infinity}] - 299266395268565444689230937067754075583779* Sum[10^(-n!), {n, 2, Infinity}] - 465234809034158007809899570315803804300290* Sum[11^(-n!), {n, 2, Infinity}] + 285887132280583324375825088698929314025198* Sum[12^(-n!), {n, 2, Infinity}] - 1455153254962389166044713884804999761815895* Sum[13^(-n!), {n, 2, Infinity}] - 433114104837923880113595542995205621923404* Sum[14^(-n!), {n, 2, Infinity}] + 1160117914615393859326075479663370802329893* Sum[15^(-n!), {n, 2, Infinity}] - 847974925978145426909219295179634996686481* Sum[16^(-n!), {n, 2, Infinity}] - 478039072552888313172646196547768919885589* Sum[17^(-n!), {n, 2, Infinity}] - 1207220432365216840785401069506724372409135* Sum[18^(-n!), {n, 2, Infinity}] - 565442430159872587593563683618559838970756* Sum[19^(-n!), {n, 2, Infinity}] + 281004860950920620115935860078699244770248* Sum[20^(-n!), {n, 2, Infinity}] + 38542239404878995475716101692658191802366* Sum[21^(-n!), {n, 2, Infinity}] + 1442550945203909873532394738550482603093249* Sum[22^(-n!), {n, 2, Infinity}] - 1070315093526954824158169515996581349047557* Sum[23^(-n!), {n, 2, Infinity}] - 1196569785451215366214358216221843708950368* Sum[24^(-n!), {n, 2, Infinity}] - 1504057272276809031352388906961647123416340* Sum[25^(-n!), {n, 2, Infinity}] + 475291034780375689238862608191953323145543* Sum[26^(-n!), {n, 2, Infinity}] - 524027840995878115513925912794868447867202* Sum[27^(-n!), {n, 2, Infinity}] - 1620441974573474591410837709104326096167725* Sum[28^(-n!), {n, 2, Infinity}] - 414214539885862686599974983033151068422599* Sum[29^(-n!), {n, 2, Infinity}] + 559985517658515157527015054675982392369751* Sum[30^(-n!), {n, 2, Infinity}] + 487621074248268136802789768671137109262588* Sum[31^(-n!), {n, 2, Infinity}] + 674666619828450995162006462639617999407097* Sum[32^(-n!), {n, 2, Infinity}] - 106618498602952986491811414739566870171557* Sum[33^(-n!), {n, 2, Infinity}] + 1168555124211830147037898399967040382528817* Sum[34^(-n!), {n, 2, Infinity}] + 85390599906882793536181196132857692177667* Sum[35^(-n!), {n, 2, Infinity}] + 569708755874101365117562354940363507455858* Sum[36^(-n!), {n, 2, Infinity}] - 435805753404222631106858418183069441499085* Sum[37^(-n!), {n, 2, Infinity}] - 100632304065941203378884812209693888341737* Sum[38^(-n!), {n, 2, Infinity}] + 175980026251533802235364343828740109328330* Sum[39^(-n!), {n, 2, Infinity}] - 203093029693440514891257626039459460579530* Sum[40^(-n!), {n, 2, Infinity}] - 527109494975226274138625409584388590824167* Sum[41^(-n!), {n, 2, Infinity}] - 1441919975136263248521894296063643037329957* Sum[42^(-n!), {n, 2, Infinity}] + 302034343438868384200878876044014773620801* Sum[43^(-n!), {n, 2, Infinity}] + 583248854885837765648122572543150789918959* Sum[44^(-n!), {n, 2, Infinity}] + 702270194902490380525021700597036955614413* Sum[45^(-n!), {n, 2, Infinity}] - 500436526244688040036598186854976926921324* Sum[46^(-n!), {n, 2, Infinity}] + 81489006025013640398934425492602897258138* Sum[47^(-n!), {n, 2, Infinity}] - 761473920288312286531262650480401117313772* Sum[48^(-n!), {n, 2, Infinity}] - 1034990444893569446002055543630756560017555* Sum[49^(-n!), {n, 2, Infinity}] - 754253538292494367301525764238807361811031* Sum[50^(-n!), {n, 2, Infinity}] - 1132795850299834123210924644846840733101239* Sum[51^(-n!), {n, 2, Infinity}] + 2387010927143877767508686859158595028391373* Sum[52^(-n!), {n, 2, Infinity}] - 1089822591277202578314471461685208924013469* Sum[53^(-n!), {n, 2, Infinity}] - 81001683977437981798666572764812171960730* Sum[54^(-n!), {n, 2, Infinity}] - 582248537457995534060569443329206686787816* Sum[55^(-n!), {n, 2, Infinity}] + 168776064302334723300910242823139170872068* Sum[56^(-n!), {n, 2, Infinity}] - 424077533292026559795307574966795337212312* Sum[57^(-n!), {n, 2, Infinity}] + 1109947769465243875535508831964972433450393* Sum[58^(-n!), {n, 2, Infinity}] + 612691804838323729773604875710797428704379* Sum[59^(-n!), {n, 2, Infinity}] - 846613643602017244023404751070377238390310* Sum[60^(-n!), {n, 2, Infinity}] - 637234398046077593684936948711478827364827* Sum[61^(-n!), {n, 2, Infinity}] - 595473419173144376910893545292289278055168* Sum[62^(-n!), {n, 2, Infinity}] - 798382236515911107795355064641213265865957* Sum[63^(-n!), {n, 2, Infinity}] - 1475710630751426353616979652407251998437441* Sum[64^(-n!), {n, 2, Infinity}] - 349266086857445890482716730556079686616657* Sum[65^(-n!), {n, 2, Infinity}] + 576297702314586207635300890940428731650513* Sum[66^(-n!), {n, 2, Infinity}] + 64424778631246518288433825435442342473108* Sum[67^(-n!), {n, 2, Infinity}] + 768355660157040286478201814231161522585435* Sum[68^(-n!), {n, 2, Infinity}] + 293710790604950219816671224005902252977446* Sum[69^(-n!), {n, 2, Infinity}] - 559797054427464244707254151567534633845493* Sum[70^(-n!), {n, 2, Infinity}] + 80950405776998237843876379200957275189067* Sum[71^(-n!), {n, 2, Infinity}] + 1670729414137120803674093465162507318808519* Sum[72^(-n!), {n, 2, Infinity}] + 775685504163875455988543406432695543816217* Sum[73^(-n!), {n, 2, Infinity}] + 1333091995709003174575758585461773893418308* Sum[74^(-n!), {n, 2, Infinity}] - 155872950652287384316462697258741573600564* Sum[75^(-n!), {n, 2, Infinity}] + 1012411988025439131263676384839491452505632* Sum[76^(-n!), {n, 2, Infinity}] + 1435991513389482138446348675629225179237248* Sum[77^(-n!), {n, 2, Infinity}] - 163117194140768456768697927667437080083367* Sum[78^(-n!), {n, 2, Infinity}] - 1894571430791935332300955306245180675566119* Sum[79^(-n!), {n, 2, Infinity}] - 88650069624075571861901184013020969336140* Sum[80^(-n!), {n, 2, Infinity}] - 2443195669827387130450675747324794999468278* Sum[81^(-n!), {n, 2, Infinity}] - 445403791149756764405424065112370443906832* Sum[82^(-n!), {n, 2, Infinity}] + 1011876040895889660018147477114342871487124* Sum[83^(-n!), {n, 2, Infinity}] - 841225667122861617814628363282657839089313* Sum[84^(-n!), {n, 2, Infinity}] + 781153505932427407408615277783615012304737* Sum[85^(-n!), {n, 2, Infinity}] - 1512669741220256920604392332262610689126748* Sum[86^(-n!), {n, 2, Infinity}] - 116806211318565735630179723946845979649289* Sum[87^(-n!), {n, 2, Infinity}] + 429130091032045229595229240522857578374793* Sum[88^(-n!), {n, 2, Infinity}] + 506765034962670771473975504854800595831172* Sum[89^(-n!), {n, 2, Infinity}] - 879274842685359858085342642938644291494265* Sum[90^(-n!), {n, 2, Infinity}] - 74882381833281682041429986677690145192329* Sum[91^(-n!), {n, 2, Infinity}] - 1481962754796105396642750584065054290318576* Sum[92^(-n!), {n, 2, Infinity}] + 371582750848581648295884969326092328244938* Sum[93^(-n!), {n, 2, Infinity}] - 548940842526326295585707606326385101045388* Sum[94^(-n!), {n, 2, Infinity}] - 214364159079515020789428860678593442975390* Sum[95^(-n!), {n, 2, Infinity}] - 792746436246894058263311867715612724830428* Sum[96^(-n!), {n, 2, Infinity}] + 98779136320327876056553220251060452284968* Sum[97^(-n!), {n, 2, Infinity}] + 420817031258715595750978255792960491285033* Sum[98^(-n!), {n, 2, Infinity}] + 133685347739575372427537403082058930459109* Sum[99^(-n!), {n, 2, Infinity}] + 624039311167296059305190395710287624563274* Sum[100^(-n!), {n, 2, Infinity}] - 1033617019679927139078081654715284819642299* Sum[101^(-n!), {n, 2, Infinity}] - 684584444208170176786209670810376971046901* Sum[102^(-n!), {n, 2, Infinity}])/ 85378145960320201007017729300346246766531, 5000]  Attachments:
 These approximations are getting too big, so I will stop here, unless I need them for something else beside cataloging them. So here is the last one, giving a teeny more that 5000 digits. The following is from the attached notebook, 5KAPPROXMRB.nb . TranscendentalRecognize[(num_)?NumericQ, (basis_)?VectorQ] := Module[{lr, ans}, lr = FindIntegerNullVector[Prepend[ N[basis, Precision[num]], num]]; ans = Rest[lr] . basis/First[lr]; Sign[N[ans]]*Sign[num]*ans] m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 5050]; Nulls = Table[Sum[x^(-n!), {n, 2, Infinity}], {x, 2, 151}]; NullTranscendentalRecognize[N[m, 5000], s[[2 ;; 150]]]Used just once in checking below.  N[m - (57731535531534074556671315066869265517* Sum[3^(-n!), {n, 2, Infinity}] - 44095020796241304068065167632121179263* Sum[4^(-n!), {n, 2, Infinity}] - 21899634945056310153254484279892470136* Sum[5^(-n!), {n, 2, Infinity}] + 64198663255948314049277738781090759487* Sum[6^(-n!), {n, 2, Infinity}] + 28871295479878266141611363045637855466* Sum[7^(-n!), {n, 2, Infinity}] + 11666986343030428973126152660286950773* Sum[8^(-n!), {n, 2, Infinity}] - 65205047843667188059323268693881332325* Sum[9^(-n!), {n, 2, Infinity}] + 75104610169275035900724312103934100381* Sum[10^(-n!), {n, 2, Infinity}] - 29659362700519526408616322083811766955* Sum[11^(-n!), {n, 2, Infinity}] + 30966513369982825469148848886354858271* Sum[12^(-n!), {n, 2, Infinity}] + 21331785169137353264577401878384599591* Sum[13^(-n!), {n, 2, Infinity}] + 3065642217622990749338289637337595091* Sum[14^(-n!), {n, 2, Infinity}] - 23383629125779495670814889192055006876* Sum[15^(-n!), {n, 2, Infinity}] - 72248291162176659416431990703859021856* Sum[16^(-n!), {n, 2, Infinity}] - 29759958353350071889162668457408175096* Sum[17^(-n!), {n, 2, Infinity}] + 100500720770089502902929450889028451472* Sum[18^(-n!), {n, 2, Infinity}] - 109739483085795430153492729799779050761* Sum[19^(-n!), {n, 2, Infinity}] + 58589474174990418993950489289592357698* Sum[20^(-n!), {n, 2, Infinity}] + 60455697244272155270448262833737276946* Sum[21^(-n!), {n, 2, Infinity}] + 31504035743595585662940887847142084870* Sum[22^(-n!), {n, 2, Infinity}] - 48428472373363769656363485066997912215* Sum[23^(-n!), {n, 2, Infinity}] - 24289904406504482178658762157762527838* Sum[24^(-n!), {n, 2, Infinity}] + 17254138305058267452388443796291524541* Sum[25^(-n!), {n, 2, Infinity}] - 25662209434405915705512396386982790445* Sum[26^(-n!), {n, 2, Infinity}] + 131732902657106848368469437016782454281* Sum[27^(-n!), {n, 2, Infinity}] + 55780100174020505028242079005886202452* Sum[28^(-n!), {n, 2, Infinity}] + 23492866909210871697083700499024822222* Sum[29^(-n!), {n, 2, Infinity}] + 3721897447313079132569450891021462070* Sum[30^(-n!), {n, 2, Infinity}] + 59331853064471643662111946758203650513* Sum[31^(-n!), {n, 2, Infinity}] + 77517119881558651969131820934219910057* Sum[32^(-n!), {n, 2, Infinity}] + 4206182612817582191667460877993504262* Sum[33^(-n!), {n, 2, Infinity}] - 5917283081874253631185572216907437926* Sum[34^(-n!), {n, 2, Infinity}] + 40361106058146017030833827947888339635* Sum[35^(-n!), {n, 2, Infinity}] + 54160435650934666371505397996930758058* Sum[36^(-n!), {n, 2, Infinity}] - 10318650648664886915018810951301369219* Sum[37^(-n!), {n, 2, Infinity}] - 21674149094241614039408992207631674559* Sum[38^(-n!), {n, 2, Infinity}] - 83632751540300288351296574455004812379* Sum[39^(-n!), {n, 2, Infinity}] + 2714345984623462589205452084791404375* Sum[40^(-n!), {n, 2, Infinity}] - 54173469179400163618760585322248571300* Sum[41^(-n!), {n, 2, Infinity}] + 3416604931787268282013119958491840497* Sum[42^(-n!), {n, 2, Infinity}] + 46706272725633962846591582479685518280* Sum[43^(-n!), {n, 2, Infinity}] - 22340740488893796575845713085665268220* Sum[44^(-n!), {n, 2, Infinity}] - 77874080439023285153613799838517359373* Sum[45^(-n!), {n, 2, Infinity}] + 79587148722657401903646568655349069434* Sum[46^(-n!), {n, 2, Infinity}] - 57299264530274322352557638096139739497* Sum[47^(-n!), {n, 2, Infinity}] - 6443167031643195030168407942967209231* Sum[48^(-n!), {n, 2, Infinity}] - 61395454494662187525617074996710546689* Sum[49^(-n!), {n, 2, Infinity}] - 11619673414316088840208290422423993752* Sum[50^(-n!), {n, 2, Infinity}] + 7015638425196477888296968310449618064* Sum[51^(-n!), {n, 2, Infinity}] + 21285977961585179173108872508295066262* Sum[52^(-n!), {n, 2, Infinity}] + 18833426693550740250200144451611433575* Sum[53^(-n!), {n, 2, Infinity}] - 52937403172753792794830709996831353297* Sum[54^(-n!), {n, 2, Infinity}] - 14867673984569243398186202710971388681* Sum[55^(-n!), {n, 2, Infinity}] - 94482443311644486128430406519155399632* Sum[56^(-n!), {n, 2, Infinity}] + 40235684381039754780167986402468093663* Sum[57^(-n!), {n, 2, Infinity}] + 91591589608109381816747858875698034947* Sum[58^(-n!), {n, 2, Infinity}] - 18293627536835752608890255749718210100* Sum[59^(-n!), {n, 2, Infinity}] - 58324966255662817852061676368317475344* Sum[60^(-n!), {n, 2, Infinity}] + 57656624502801667057834909431037192717* Sum[61^(-n!), {n, 2, Infinity}] + 20935616850996435879546270576024167019* Sum[62^(-n!), {n, 2, Infinity}] + 18451216415057205247641143251017373760* Sum[63^(-n!), {n, 2, Infinity}] + 85808867747485942085772910625263787953* Sum[64^(-n!), {n, 2, Infinity}] + 71550751826275229063345444850146128320* Sum[65^(-n!), {n, 2, Infinity}] + 16127347579828036974574753798828507872* Sum[66^(-n!), {n, 2, Infinity}] - 8483005071033316725549832707304227990* Sum[67^(-n!), {n, 2, Infinity}] + 69647119603654268083221111968347248441* Sum[68^(-n!), {n, 2, Infinity}] + 33580943378314302118381767680050837605* Sum[69^(-n!), {n, 2, Infinity}] + 17027657955814127839286423878196624957* Sum[70^(-n!), {n, 2, Infinity}] - 38311102715841846211801690604531307536* Sum[71^(-n!), {n, 2, Infinity}] - 63189103282054975017269674235313560905* Sum[72^(-n!), {n, 2, Infinity}] - 49164273883130375853898730519798432277* Sum[73^(-n!), {n, 2, Infinity}] - 31753164586977509575224959809384829639* Sum[74^(-n!), {n, 2, Infinity}] + 13172438190297907801779959545441005911* Sum[75^(-n!), {n, 2, Infinity}] - 13311538012642998140206646875084032906* Sum[76^(-n!), {n, 2, Infinity}] - 3594703844561927312953730401527983068* Sum[77^(-n!), {n, 2, Infinity}] - 71212752776422086409455238372506995961* Sum[78^(-n!), {n, 2, Infinity}] + 46270051330640786710165883133853991246* Sum[79^(-n!), {n, 2, Infinity}] + 128435509337369614063672322025790317821* Sum[80^(-n!), {n, 2, Infinity}] + 2284155794720492254077614563223123089* Sum[81^(-n!), {n, 2, Infinity}] - 20454605090757316455784842163150930202* Sum[82^(-n!), {n, 2, Infinity}] + 15269572842729141817229302478539842818* Sum[83^(-n!), {n, 2, Infinity}] - 67853651799682565921858759245378860029* Sum[84^(-n!), {n, 2, Infinity}] - 7420016241806218445084306897539407120* Sum[85^(-n!), {n, 2, Infinity}] + 14995960344272264911026064897057582228* Sum[86^(-n!), {n, 2, Infinity}] - 46508210131269373298559574055788132440* Sum[87^(-n!), {n, 2, Infinity}] - 20827116575862053194096940132387721994* Sum[88^(-n!), {n, 2, Infinity}] - 34901546305911194024567186981012437606* Sum[89^(-n!), {n, 2, Infinity}] + 80723093887038779483313385645769643500* Sum[90^(-n!), {n, 2, Infinity}] + 91298015023732805705443828408804302475* Sum[91^(-n!), {n, 2, Infinity}] + 81914149461025379900990501612006241152* Sum[92^(-n!), {n, 2, Infinity}] - 83022867179955413506823990831899646978* Sum[93^(-n!), {n, 2, Infinity}] - 19825217698560073963597842251901616257* Sum[94^(-n!), {n, 2, Infinity}] + 3222466975927586221463692362305204380* Sum[95^(-n!), {n, 2, Infinity}] + 62432261944374190390363726971924682556* Sum[96^(-n!), {n, 2, Infinity}] - 1307205745820583693824658255976746779* Sum[97^(-n!), {n, 2, Infinity}] + 28546553867155733246796355632159408964* Sum[98^(-n!), {n, 2, Infinity}] + 28891921543305070619215026866436199044* Sum[99^(-n!), {n, 2, Infinity}] + 854726094389218310649167628752870231* Sum[100^(-n!), {n, 2, Infinity}] - 14763391439605645054369729804861276095* Sum[101^(-n!), {n, 2, Infinity}] - 15373446902497158633176739257794930459* Sum[102^(-n!), {n, 2, Infinity}] - 120798493462125307308655930369628726675* Sum[103^(-n!), {n, 2, Infinity}] + 52469138834626960041562563946768572181* Sum[104^(-n!), {n, 2, Infinity}] - 6611709288516320545870471975220980889* Sum[105^(-n!), {n, 2, Infinity}] + 1867398543543123228627894767305534096* Sum[106^(-n!), {n, 2, Infinity}] - 41120969884927540527684937142042509555* Sum[107^(-n!), {n, 2, Infinity}] + 24714859574703194917646384967949359756* Sum[108^(-n!), {n, 2, Infinity}] + 21406960988304903266507168441981141721* Sum[109^(-n!), {n, 2, Infinity}] + 25038835436090437233406109185855450255* Sum[110^(-n!), {n, 2, Infinity}] - 59828824335358169580429717457287320545* Sum[111^(-n!), {n, 2, Infinity}] + 154716505347217330839630246672590287370* Sum[112^(-n!), {n, 2, Infinity}] - 14008249603244862952574536874073307143* Sum[113^(-n!), {n, 2, Infinity}] - 60092773884925890446958593479837943494* Sum[114^(-n!), {n, 2, Infinity}] - 21004274181981080897701041545003036963* Sum[115^(-n!), {n, 2, Infinity}] + 111223157013811580104704722696930875247* Sum[116^(-n!), {n, 2, Infinity}] - 131096210353465154292178973243593655894* Sum[117^(-n!), {n, 2, Infinity}] - 100163463831512802687978802486240031353* Sum[118^(-n!), {n, 2, Infinity}] + 57595647530931103298163259136147823663* Sum[119^(-n!), {n, 2, Infinity}] - 10358776687309523297047179676726518045* Sum[120^(-n!), {n, 2, Infinity}] + 38302900634278996392516308092330716712* Sum[121^(-n!), {n, 2, Infinity}] + 47804000025054638274936539029104594109* Sum[122^(-n!), {n, 2, Infinity}] + 4700122543740534458043165957297794992* Sum[123^(-n!), {n, 2, Infinity}] - 57246490768146036272392003474414317632* Sum[124^(-n!), {n, 2, Infinity}] - 101895023006956616686134883219546794163* Sum[125^(-n!), {n, 2, Infinity}] - 94556932576515823590049207532045378601* Sum[126^(-n!), {n, 2, Infinity}] - 703128403991430321160538072784015573* Sum[127^(-n!), {n, 2, Infinity}] + 22782338374148948514085790694781879236* Sum[128^(-n!), {n, 2, Infinity}] + 73771246494037944157479555600739327812* Sum[129^(-n!), {n, 2, Infinity}] - 4553058762763582479505833213730159831* Sum[130^(-n!), {n, 2, Infinity}] + 3741392875646463440545717580776717213* Sum[131^(-n!), {n, 2, Infinity}] + 9487508228156559427322012324493744538* Sum[132^(-n!), {n, 2, Infinity}] + 36655109112750606669557335936705614924* Sum[133^(-n!), {n, 2, Infinity}] - 20092593206525648655881369770537750374* Sum[134^(-n!), {n, 2, Infinity}] - 9592512922811837810310769163219047391* Sum[135^(-n!), {n, 2, Infinity}] - 98799727244199015713460035512968035899* Sum[136^(-n!), {n, 2, Infinity}] - 30110420637800073581700420185763072042* Sum[137^(-n!), {n, 2, Infinity}] - 54354156218806994727210229408530239149* Sum[138^(-n!), {n, 2, Infinity}] + 73230976379721544869544476715993132377* Sum[139^(-n!), {n, 2, Infinity}] - 95643543550167132551902068508307710657* Sum[140^(-n!), {n, 2, Infinity}] - 36652120470322008922568234491289311890* Sum[141^(-n!), {n, 2, Infinity}] + 36624482932592290727011693355435008314* Sum[142^(-n!), {n, 2, Infinity}] - 45572259185730347719999417604003835132* Sum[143^(-n!), {n, 2, Infinity}] - 10388528564290377184185465655883250949* Sum[144^(-n!), {n, 2, Infinity}] + 11058524828785980139984714279831742890* Sum[145^(-n!), {n, 2, Infinity}] - 81594137033369586975137123509511783441* Sum[146^(-n!), {n, 2, Infinity}] - 19543355984406037564448478952611709118* Sum[147^(-n!), {n, 2, Infinity}] - 109525526881866971076008622439807099934* Sum[148^(-n!), {n, 2, Infinity}] + 16300270168294057208852874105352647785* Sum[149^(-n!), {n, 2, Infinity}] - 29238726445163807704491144908413171592* Sum[150^(-n!), {n, 2, Infinity}] + 8312704803526195122769807573601867172* Sum[151^(-n!), {n, 2, Infinity}])/ 29555894016645948876604458335734030343, 5050] gives 1.45472134483986690932420586937649500231787...*^-5003Also a notebook with a 6000 digit approx is below and attached. I Decided to add it to see if anyone can figure out a pattern. (315981424734279695944988454735349008* Sum[3^(-n!), {n, 2, Infinity}] + 263423996018012329771736458181315695* Sum[4^(-n!), {n, 2, Infinity}] - 225089838855827096167206491226890453* Sum[5^(-n!), {n, 2, Infinity}] + 79001337992367787044034193000391921* Sum[6^(-n!), {n, 2, Infinity}] + 82678726909932185622246011042919182* Sum[7^(-n!), {n, 2, Infinity}] - 247512322536610252151262924731282435* Sum[8^(-n!), {n, 2, Infinity}] - 373025156916679117399944255380192973* Sum[9^(-n!), {n, 2, Infinity}] + 489448126438191836401630086136736784* Sum[10^(-n!), {n, 2, Infinity}] - 38974476779796180723370173855027274* Sum[11^(-n!), {n, 2, Infinity}] + 302228585683016982083843206903263152* Sum[12^(-n!), {n, 2, Infinity}] - 509342246318522665519734352063206162* Sum[13^(-n!), {n, 2, Infinity}] - 85350690530786302144573962214621151* Sum[14^(-n!), {n, 2, Infinity}] - 479351485358820743688775530832165702* Sum[15^(-n!), {n, 2, Infinity}] + 204477791516638635965393390265208484* Sum[16^(-n!), {n, 2, Infinity}] + 13385289748029332545946019740090415* Sum[17^(-n!), {n, 2, Infinity}] - 157087956179230468329160163646308887* Sum[18^(-n!), {n, 2, Infinity}] + 263474578355711774585051601431349774* Sum[19^(-n!), {n, 2, Infinity}] + 97689520253762483063496321133985796* Sum[20^(-n!), {n, 2, Infinity}] - 113195866425880453454502398064093977* Sum[21^(-n!), {n, 2, Infinity}] + 174716639458316549154287353779689292* Sum[22^(-n!), {n, 2, Infinity}] + 217824330687208262192110074949241343* Sum[23^(-n!), {n, 2, Infinity}] - 7942668367405234213472069233019886* Sum[24^(-n!), {n, 2, Infinity}] - 207393236117386950046340580124345419* Sum[25^(-n!), {n, 2, Infinity}] + 172383688497561861995251147333700938* Sum[26^(-n!), {n, 2, Infinity}] + 144408938241864702564745188333518242* Sum[27^(-n!), {n, 2, Infinity}] + 19856450607914676901940694502412091* Sum[28^(-n!), {n, 2, Infinity}] + 71827614351161963367675827441301274* Sum[29^(-n!), {n, 2, Infinity}] + 209883409286900104613955679964303833* Sum[30^(-n!), {n, 2, Infinity}] + 91110178999390077428396192426673845* Sum[31^(-n!), {n, 2, Infinity}] - 119884940466936964776666254085163196* Sum[32^(-n!), {n, 2, Infinity}] + 51371406160894613525692832024139738* Sum[33^(-n!), {n, 2, Infinity}] - 60470663699938201840915277998362143* Sum[34^(-n!), {n, 2, Infinity}] - 26379597856446288295640917038864490* Sum[35^(-n!), {n, 2, Infinity}] - 252578964335856408098941641365904628* Sum[36^(-n!), {n, 2, Infinity}] + 313297948156460384849465701023469*Sum[37^(-n!), {n, 2, Infinity}] - 238465182471529245716545180075648340* Sum[38^(-n!), {n, 2, Infinity}] - 310571320016173268427612358922750295* Sum[39^(-n!), {n, 2, Infinity}] + 438570535340392744689031167389555374* Sum[40^(-n!), {n, 2, Infinity}] + 148973823070752413601420185118514385* Sum[41^(-n!), {n, 2, Infinity}] - 12933064459296489356215220491969905* Sum[42^(-n!), {n, 2, Infinity}] + 125229095153406272804498756316486543* Sum[43^(-n!), {n, 2, Infinity}] + 179919250855304262282694440172008701* Sum[44^(-n!), {n, 2, Infinity}] - 65364651686252682986073857154785133* Sum[45^(-n!), {n, 2, Infinity}] - 94847344263989779988112499508155948* Sum[46^(-n!), {n, 2, Infinity}] - 621627953356444169434114635659571781* Sum[47^(-n!), {n, 2, Infinity}] + 241831496547243211024459553012456108* Sum[48^(-n!), {n, 2, Infinity}] - 80382601092110385430867421360994192* Sum[49^(-n!), {n, 2, Infinity}] + 214319245669088348481923244047278353* Sum[50^(-n!), {n, 2, Infinity}] + 263616311677846637171116951814248834* Sum[51^(-n!), {n, 2, Infinity}] - 47217446846628628172714044765259562* Sum[52^(-n!), {n, 2, Infinity}] - 160559943742761005221065628056915360* Sum[53^(-n!), {n, 2, Infinity}] + 124288679032449106926938304339538588* Sum[54^(-n!), {n, 2, Infinity}] - 197937587153815754792688590868006073* Sum[55^(-n!), {n, 2, Infinity}] + 54557603465371360054436979408777625* Sum[56^(-n!), {n, 2, Infinity}] - 97162833540442963477079325054823592* Sum[57^(-n!), {n, 2, Infinity}] - 90320339434632806839828901977497508* Sum[58^(-n!), {n, 2, Infinity}] - 15869448917653025921598376361231822* Sum[59^(-n!), {n, 2, Infinity}] + 97944774418250015033878296784456445* Sum[60^(-n!), {n, 2, Infinity}] + 36798433281590416847750764293609436* Sum[61^(-n!), {n, 2, Infinity}] + 252639830095148671730012461570230583* Sum[62^(-n!), {n, 2, Infinity}] - 60454733566848343419447448832119054* Sum[63^(-n!), {n, 2, Infinity}] + 412531740074892182558565222439091209* Sum[64^(-n!), {n, 2, Infinity}] + 222186389489693893971771846666428316* Sum[65^(-n!), {n, 2, Infinity}] + 152444052645393490345585279197665837* Sum[66^(-n!), {n, 2, Infinity}] - 45712465705543638455056885162906980* Sum[67^(-n!), {n, 2, Infinity}] + 537888021247460244376707283556926677* Sum[68^(-n!), {n, 2, Infinity}] + 41729702289089382012946311545939513* Sum[69^(-n!), {n, 2, Infinity}] + 269839406669060930667191565937804546* Sum[70^(-n!), {n, 2, Infinity}] + 135086788301021891711964442036630427* Sum[71^(-n!), {n, 2, Infinity}] - 28888022019045539583358799776076932* Sum[72^(-n!), {n, 2, Infinity}] + 113532956776247778469719191157543773* Sum[73^(-n!), {n, 2, Infinity}] - 89307745544685357903676017016484787* Sum[74^(-n!), {n, 2, Infinity}] + 47838743041867005632039446217738151* Sum[75^(-n!), {n, 2, Infinity}] + 184083250574080586193876701395692284* Sum[76^(-n!), {n, 2, Infinity}] - 135857254885547479195698346359609379* Sum[77^(-n!), {n, 2, Infinity}] - 117608774311099026843625475716325491* Sum[78^(-n!), {n, 2, Infinity}] - 73263873196621702272542398429997287* Sum[79^(-n!), {n, 2, Infinity}] - 36674398326493497474335279956752119* Sum[80^(-n!), {n, 2, Infinity}] + 1349446953914528982825041795604197* Sum[81^(-n!), {n, 2, Infinity}] - 49318698626911631983193408221702888* Sum[82^(-n!), {n, 2, Infinity}] + 312434270346509621900655479125952464* Sum[83^(-n!), {n, 2, Infinity}] + 20361467111872699753180246042732586* Sum[84^(-n!), {n, 2, Infinity}] + 37585893876831895828817156870674991* Sum[85^(-n!), {n, 2, Infinity}] - 92324616850053384287711048884039538* Sum[86^(-n!), {n, 2, Infinity}] - 12181874329276651146548526373957738* Sum[87^(-n!), {n, 2, Infinity}] + 76250223084500704962391681263286831* Sum[88^(-n!), {n, 2, Infinity}] + 174201681256909307459523915152881457* Sum[89^(-n!), {n, 2, Infinity}] - 23961565519484663694162413453690203* Sum[90^(-n!), {n, 2, Infinity}] - 149999380951599562551754971549446591* Sum[91^(-n!), {n, 2, Infinity}] + 54943757923779838427102559784033199* Sum[92^(-n!), {n, 2, Infinity}] - 8339247753293270311605791564177575* Sum[93^(-n!), {n, 2, Infinity}] + 237072493457174281387274575949807591* Sum[94^(-n!), {n, 2, Infinity}] - 259571743147388275774214807165075384* Sum[95^(-n!), {n, 2, Infinity}] + 88685372064249400413234600210246056* Sum[96^(-n!), {n, 2, Infinity}] + 51564239309247760026781789145118667* Sum[97^(-n!), {n, 2, Infinity}] + 9488491149572121359149972552544321* Sum[98^(-n!), {n, 2, Infinity}] - 100165938874819370241719527413349478* Sum[99^(-n!), {n, 2, Infinity}] - 207195325384481749820847136464019828* Sum[100^(-n!), {n, 2, Infinity}] - 54481457817463396570634375163914993* Sum[101^(-n!), {n, 2, Infinity}] + 249079767298561005825683297640400784* Sum[102^(-n!), {n, 2, Infinity}] + 29304706256650991339784984234383998* Sum[103^(-n!), {n, 2, Infinity}] + 174735356467887097758558215245744513* Sum[104^(-n!), {n, 2, Infinity}] - 108889550843958974521933476640338838* Sum[105^(-n!), {n, 2, Infinity}] + 87462022299835631237720920878276644* Sum[106^(-n!), {n, 2, Infinity}] + 352764652159751665182743140743797835* Sum[107^(-n!), {n, 2, Infinity}] - 39527540676546427988382515109034788* Sum[108^(-n!), {n, 2, Infinity}] - 197535107893227221684343578698597562* Sum[109^(-n!), {n, 2, Infinity}] + 408966239657680369638120674175630782* Sum[110^(-n!), {n, 2, Infinity}] - 179041538282929746452523277369969046* Sum[111^(-n!), {n, 2, Infinity}] - 418987347443356698579370926729452317* Sum[112^(-n!), {n, 2, Infinity}] + 212673413170115056993488568648538628* Sum[113^(-n!), {n, 2, Infinity}] + 109402498909040751573711220405298243* Sum[114^(-n!), {n, 2, Infinity}] - 251694051940381172374491380867070131* Sum[115^(-n!), {n, 2, Infinity}] + 82922590579916575323370846170661625* Sum[116^(-n!), {n, 2, Infinity}] + 141185801698917717457812216215148958* Sum[117^(-n!), {n, 2, Infinity}] + 181134491265141326525541898931448362* Sum[118^(-n!), {n, 2, Infinity}] - 132213080778822164157827259847107066* Sum[119^(-n!), {n, 2, Infinity}] - 88473636543082192730804664586530929* Sum[120^(-n!), {n, 2, Infinity}] + 301714649948755394226628716785024218* Sum[121^(-n!), {n, 2, Infinity}] + 44123179305489846774208377585409491* Sum[122^(-n!), {n, 2, Infinity}] - 164379875589719216127688610966998146* Sum[123^(-n!), {n, 2, Infinity}] + 501046982451636338386607075354653779* Sum[124^(-n!), {n, 2, Infinity}] - 364279243815675896519524672723097728* Sum[125^(-n!), {n, 2, Infinity}] - 2375527346715611768257295450807400* Sum[126^(-n!), {n, 2, Infinity}] - 19452928739717897182210893690092571* Sum[127^(-n!), {n, 2, Infinity}] + 454151299786283395540780210700230746* Sum[128^(-n!), {n, 2, Infinity}] + 37376114856713291341212108398605576* Sum[129^(-n!), {n, 2, Infinity}] + 231392269880587788448395728678209628* Sum[130^(-n!), {n, 2, Infinity}] + 50122247967226209202475595982658046* Sum[131^(-n!), {n, 2, Infinity}] + 252814970176124257483586481088928331* Sum[132^(-n!), {n, 2, Infinity}] + 77716463783359732887151339490831649* Sum[133^(-n!), {n, 2, Infinity}] - 72380011819287630601133689824982104* Sum[134^(-n!), {n, 2, Infinity}] - 197314297374390342405502640443415525* Sum[135^(-n!), {n, 2, Infinity}] - 239590676709662470841507207197013339* Sum[136^(-n!), {n, 2, Infinity}] + 95321496680663831442416429419228816* Sum[137^(-n!), {n, 2, Infinity}] - 56918591225412742784774111833641989* Sum[138^(-n!), {n, 2, Infinity}] + 112956161872387819978577025773587680* Sum[139^(-n!), {n, 2, Infinity}] - 162129715688970577971100354279299458* Sum[140^(-n!), {n, 2, Infinity}] - 165788373979348152041647621850755045* Sum[141^(-n!), {n, 2, Infinity}] + 193488202673646850093731186451320576* Sum[142^(-n!), {n, 2, Infinity}] - 548049534879807499695788178142775711* Sum[143^(-n!), {n, 2, Infinity}] + 149929900735886016561828675677323879* Sum[144^(-n!), {n, 2, Infinity}] + 109030802103795227201000545137622493* Sum[145^(-n!), {n, 2, Infinity}] + 116443234746584509377150436003118677* Sum[146^(-n!), {n, 2, Infinity}] + 7516677443054165519672701712699614* Sum[147^(-n!), {n, 2, Infinity}] + 185797817457724013365392879403492576* Sum[148^(-n!), {n, 2, Infinity}] + 163019019600562168777865146873566915* Sum[149^(-n!), {n, 2, Infinity}] + 174744583076795521301304310285603543* Sum[150^(-n!), {n, 2, Infinity}] - 5530094694507477516902247185091367* Sum[151^(-n!), {n, 2, Infinity}] - 159993283754404572876061149387146073* Sum[152^(-n!), {n, 2, Infinity}] - 20190979748366644843546991850900499* Sum[153^(-n!), {n, 2, Infinity}] - 315746713840332783648739349851450877* Sum[154^(-n!), {n, 2, Infinity}] - 220553072075068314620207647275747831* Sum[155^(-n!), {n, 2, Infinity}] - 74835040603287483205307521237225642* Sum[156^(-n!), {n, 2, Infinity}] - 11261942604143666747079961031082151* Sum[157^(-n!), {n, 2, Infinity}] + 122500315722834955403769173398284189* Sum[158^(-n!), {n, 2, Infinity}] + 19355534961471081068328784791822556* Sum[159^(-n!), {n, 2, Infinity}] - 24171664786198281859045158156631504* Sum[160^(-n!), {n, 2, Infinity}] - 37304159392600733806142005936112613* Sum[161^(-n!), {n, 2, Infinity}] - 43390526037396451624214748860917124* Sum[162^(-n!), {n, 2, Infinity}] - 178124789687088784047367909558868832* Sum[163^(-n!), {n, 2, Infinity}] - 129697771675713291186708604185038568* Sum[164^(-n!), {n, 2, Infinity}] - 287783212860743244691881417066300239* Sum[165^(-n!), {n, 2, Infinity}] - 39004936231977132468790119762278629* Sum[166^(-n!), {n, 2, Infinity}] + 40460636968805665607015035105957795* Sum[167^(-n!), {n, 2, Infinity}] - 2745798669997723668891497173903991* Sum[168^(-n!), {n, 2, Infinity}] + 15702921931594188322311337836282248* Sum[169^(-n!), {n, 2, Infinity}] + 198235069860788583685403985742163386* Sum[170^(-n!), {n, 2, Infinity}] + 164479301068157371692587032898594891* Sum[171^(-n!), {n, 2, Infinity}] + 168271290637022192246652999496409154* Sum[172^(-n!), {n, 2, Infinity}] - 28847077794524689480429578762761829* Sum[173^(-n!), {n, 2, Infinity}] + 54321502221197203755000146166549989* Sum[174^(-n!), {n, 2, Infinity}] + 74067107694636621233517904653151316* Sum[175^(-n!), {n, 2, Infinity}] - 94585072117290818703280407462082597* Sum[176^(-n!), {n, 2, Infinity}] - 305632900916807059676371405142479061* Sum[177^(-n!), {n, 2, Infinity}] + 160670158321978466344142594662162328* Sum[178^(-n!), {n, 2, Infinity}] - 132820591314239340106716063373992490* Sum[179^(-n!), {n, 2, Infinity}] + 43759078377033606010617996242164936* Sum[180^(-n!), {n, 2, Infinity}] - 26831797137219681958324963901360854* Sum[181^(-n!), {n, 2, Infinity}] + 201822390541950284764187054439769695* Sum[182^(-n!), {n, 2, Infinity}] + 28157456911833749608335322048487096* Sum[183^(-n!), {n, 2, Infinity}] - 160051393351323301197976304332905792* Sum[184^(-n!), {n, 2, Infinity}] + 42921313165203287148344700798275115* Sum[185^(-n!), {n, 2, Infinity}] + 161438923210709048939689495607283470* Sum[186^(-n!), {n, 2, Infinity}] - 236043714494690607442155545996259754* Sum[187^(-n!), {n, 2, Infinity}] - 204446821520358652220849952968433259* Sum[188^(-n!), {n, 2, Infinity}] - 265264067031194835058399060984376642* Sum[189^(-n!), {n, 2, Infinity}] - 359546925325277487563947112101078671* Sum[190^(-n!), {n, 2, Infinity}] - 79700412680229831564816575567347459* Sum[191^(-n!), {n, 2, Infinity}] - 61063651607412749537518874562893443* Sum[192^(-n!), {n, 2, Infinity}] - 226913071177053988439271980478321049* Sum[193^(-n!), {n, 2, Infinity}] - 373312029979815591768749834849100993* Sum[194^(-n!), {n, 2, Infinity}] + 92136834648577784511024142330126493* Sum[195^(-n!), {n, 2, Infinity}] + 380979676527824189056642832302506663* Sum[196^(-n!), {n, 2, Infinity}] + 162056853965324307731243153199521550* Sum[197^(-n!), {n, 2, Infinity}] + 142623707639267739602457347131389959* Sum[198^(-n!), {n, 2, Infinity}] - 63536206327935361530376288963445714* Sum[199^(-n!), {n, 2, Infinity}] + 324533006813555086981415178554954199* Sum[200^(-n!), {n, 2, Infinity}] + 74641094495220261486525282033370447* Sum[201^(-n!), {n, 2, Infinity}])/ 223954053633340275635823197718054367 ` Attachments: