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Mathematics Wolfram Language Numerical Computation

The EulerGamma*Log[2] - Log[2]^2/2 and 5 (Log[Pi] - Log[2]) Problems

Marvin Ray Burns

Marvin Ray Burns, original investigator of the MRB constant

Posted 9 years ago

Considering the MRB constant which is approximated by NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}, WorkingPrecision -> 70, Method -> "AlternatingSigns"] giving 0.18785964246206712024851793405427323005590309490013878617200468547342, I wondered what happens when we change 1/n in n^(1/n) to 1 over other functions involving n? Then I figured out that Table[NSum[(-1)^n (1/n^(1/(n10^x)) - 1), {n, 1, Infinity}, WorkingPrecision -> 70, Method -> "AlternatingSigns"] + (EulerGammaLog[2] - Log[2]^2/2)10^-x, {x, 1, 30}] and also Table[NSum[(-1)^n (n^(1/(n10^x)) - 1), {n, 1, Infinity}, WorkingPrecision -> 70, Method -> "AlternatingSigns"] - (EulerGammaLog[2] - Log[2]^2/2)10^-x, {x, 1, 30}] mantissa's converge to the same constant even though one formula uses the reciprocal of what the other uses! {0.0002544931414649952541958458971418195836265633470373523204886572940\ 13, 2.5213764023532449457379170196206807316529005454922927744915334919\ 10^-6, 2.\ 51903843034426380209825806169280958181176781202359420358134443410^-8, 2.5188048074204347074530949670489009342955128826334903848060022510^\ -10, 2.5187814468698119802633255716680901574100656515276061645434502\ 10^-12, 2.\ 51877911083216629932053333834014449754031226765360715410821210^-14, 2.5187788772285758961340155189573949702382751156791710339624010^-16, 2.518778853868218597463431355227310369824199412932791989009110^-18, 2.51877885153218288501285260504101368177245518420697979778210^-20, 2.5187788512985793139419595256742558066734840416128903011810^-22, 2.518778851275218956836611865693088741781247050654497030310^-24, 2.51877885127288292112609451617452611252288101206067286510^-26, 2.5187788512726493175550429553874653893593576659129301510^-28, 2.518778851272625957197937801050407272439618468568710510^-30, 2.51877885127262362116222728563411794030160967033495810^-32, 2.5187788512726233875586562340926631718833484419494210^-34, 2.518778851272623364198299128938519436689477727952510^-36, 2.51877885127262336186226341842310508058657033392410^-38, 2.5187788512726233616286598473715636451504456241210^-40, 2.518778851272623361605299490266409501608587141710^-42, 2.51877885127262336160296345455589408725454211610^-44, 2.5187788512726233616027298509848425458203718510^-46, 2.518778851272623361602706490627737391689295410^-48, 2.51877885127262336160270415459202687639959410^-50, 2.5187788512726233616027039209884558261046810^-52, 2.518778851272623361602703897628098733415810^-54, 2.51877885127262336160270389529206314755310^-56, 2.5187788512726233616027038950584608230310^-58, 2.518778851272623361602703895035112931210^-60, 2.51877885127262336160270389503290154810^-62}. This is for the same reason that NSum[(-1)^n (-1 + 1/n^(10^-x/n) + (10^-x Log[n])/n), {n, 1, Infinity}, WorkingPrecision -> 70, Method -> "AlternatingSigns"] /. x -> 30 and NSum[(-1)^n (-1 + n^(10^-x/n) - (10^-x Log[n])/n), {n, 1, Infinity}, WorkingPrecision -> 70, Method -> "AlternatingSigns"] /. x -> 30 give 2.51877885127262336160270389503276304518288666816092953082281396924871\ 10^-62 (For version 3.1 you will have to use x->27 or less.) (I wonder what closed form starts with 2.518778851272623361602703895?) While changing the sign of "1:" NSum[(-1)^n (1 + n^(10^-x/n) - (10^-x Log[n])/n), {n, 1, Infinity}, WorkingPrecision -> 70, Method -> "AlternatingSigns"] /. x -> 30 gives a "divergent sum" of -1.0000000000000000000000000000000000000000000000000000000000038475945 And another constant, with formulas behaving differently, I noticed is from Table[NSum[(-1)^n (n^(1/(n + 10^x)) - 1), {n, 1, Infinity}, WorkingPrecision -> 70, Method -> "AlternatingSigns"] - (5 (Log[Pi] - Log[2]))10^-(x + 1), {x, 1, 30}] which gives {-0.002158265971057576296292446726899637434277905244619602754711500291\ 280, -0.00002326878830432348151050186189530613369186033070715442565971\ 5184159, -2.\ 34497249854041366905115722694391433766118268237812126559372821110^-7,\ -2.3467953041108743185166059751383162918387750898356477082447408110^\ -9, -2.3469777203157053575942148177302233368868838187419725158605417\ 10^-11, -2.\ 34699596329394126061023648686282511473984998720931014014684810^-13, \ -2.3469977876053436553262541449945615160496084863294484805695710^-15,\ -2.346997970036619684118570769559395597449215581220894871237810^-17,\ -2.34699798827974864489228616604407001762631294619079307588110^-19, \ -2.3469979901040615545486038196197337250541261257921467947610^-21, \ -2.346997990286492845650025047393159120174360305871916103110^-23, \ -2.34699799030473597476152506479593683700661841046116861610^-25, \ -2.3469979903065602876726886454824702370501245034643317110^-27, \ -2.346997990306742718963805139340586134614665012071864710^-29, \ -2.34699799030676096209291679008429234994799773300946410^-31, \ -2.3469979903067627864058279551722419177371029084478510^-33, \ -2.346997990306762968837119071681172663978589546869610^-35, \ -2.34699799030676298708024818333206709649754797760010^-37, \ -2.3469979903067629889045610944971565533302319751210^-39, \ -2.346997990306762989086992385613665499167708824210^-41, \ -2.34699799030676298910523551472531639393700427210^-43, \ -2.3469979903067629891070598276364814852558460810^-45, \ -2.346997990306762989107242258927598012806717210^-47, \ -2.34699799030676298910726050205670984975167210^-49, \ -2.3469979903067629891072623263696228753448510^-51, \ -2.346997990306762989107262508800932596891010^-53, \ -2.34699799030676298910726252704424775891410^-55, \ -2.3469979903067629891072625288704211738010^-57, \ -2.346997990306762989107262529071457502110^-59, \ -2.34699799030676298910726252927575100310^-61} (Likewise I wonder what closed form starts with 2.34699799030676298910726?) However here we have a different constant converged to by using the reciprocal Table[NSum[(-1)^n (1/n^(1/(n + 10^x)) - 1), {n, 1, Infinity}, WorkingPrecision -> 70, Method -> "AlternatingSigns"] + (5 (Log[Pi] - Log[2]))10^-(x + 1), {x, 1, 30}] giving {0.0027281794786312486595717428387460304770563641777457765720218456040\ 40, 0.0000293550399240151127322722379565607620916546859019350100125647\ 09573, 2.9551297193773754366885564592734651504693514196068487319926172\ 9610^-7, 2.9570736539515365582615546184716902042921506862134750212997159910^-\ 9, 2.957267852560542606352981411445828264222773278505446549105694310^\ -11, 2.957287270464422416597614464460409791592623556524877730647258\ 10^-13, 2.95728921223523164631517763738252605852113445125380329353929\ 10^-15, 2.9572894064121167732298398914267611418621574492035860745636\ 10^-17, 2.957289425829803327952191084785434676290469048265224666794\ 10^-19, 2.9572894277715719638447265097027964186914214236148816810010^\ -21, 2.957289427965748827238183046706248825228620779808929391610^-23, 2.95728942798516651357557073034316712942737119162985589210^-25, 2.9572894279871082822092899190062161463839119640403374310^-27, 2.957289427987302459072661642075514611400833931755382810^-29, 2.95728942798732187675899881242447439352719489265200010^-31, 2.9572894279873238185276325294397906710960709723409110^-33, 2.957289427987324012704495901141126501846539372169510^-35, 2.95728942798732403212218223831125812695170602574110^-37, 2.9572894279873240340639508720282712698843639460210^-39, 2.957289427987324034258127735399972584000251718410^-41, 2.95728942798732403427754542173714271559407239410^-43, 2.9572894279873240342794871903708597305953335610^-45, 2.957289427987324034279681367234231450514446310^-47, 2.95728942798732403427970078492056880669622610^-49, 2.9572894279873240342797027266892043842130910^-51, 2.957289427987324034279702920866086360951610^-53, 2.95728942798732403427970294028395874849410^-55, 2.9572894279873240342797029422275878859310^-57, 2.957289427987324034279702942440369786510^-59, 2.95728942798732403427970294264583784510^-61} I could also ask about the constant 2.957289427987324034279702942.

Considering the MRB constant which is approximated by

NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}, WorkingPrecision -> 70, 
 Method -> "AlternatingSigns"]

giving

0.18785964246206712024851793405427323005590309490013878617200468547342,

I wondered what happens when we change 1/n in n^(1/n) to 1 over other functions involving n?

Then I figured out that

Table[NSum[(-1)^n (1/n^(1/(n*10^x)) - 1), {n, 1, Infinity}, 
   WorkingPrecision -> 70, 
   Method -> 
    "AlternatingSigns"] + (EulerGamma*Log[2] - Log[2]^2/2)*10^-x, {x, 
  1, 30}]

and also

Table[NSum[(-1)^n (n^(1/(n*10^x)) - 1), {n, 1, Infinity}, 
   WorkingPrecision -> 70, 
   Method -> 
    "AlternatingSigns"] - (EulerGamma*Log[2] - Log[2]^2/2)*10^-x, {x, 
  1, 30}]

mantissa's converge to the same constant even though one formula uses the reciprocal of what the other uses!

 {0.0002544931414649952541958458971418195836265633470373523204886572940\
 13, 2.5213764023532449457379170196206807316529005454922927744915334919\
 *10^-6, 2.\
 519038430344263802098258061692809581811767812023594203581344434*10^-8,
   2.51880480742043470745309496704890093429551288263349038480600225*10^\
 -10, 2.5187814468698119802633255716680901574100656515276061645434502*\
 10^-12, 2.\
 518779110832166299320533338340144497540312267653607154108212*10^-14, 
  2.51877887722857589613401551895739497023827511567917103396240*10^-16,
   2.5187788538682185974634313552273103698241994129327919890091*10^-18,
   2.518778851532182885012852605041013681772455184206979797782*10^-20, 
  2.51877885129857931394195952567425580667348404161289030118*10^-22, 
  2.5187788512752189568366118656930887417812470506544970303*10^-24, 
  2.518778851272882921126094516174526112522881012060672865*10^-26, 
  2.51877885127264931755504295538746538935935766591293015*10^-28, 
  2.5187788512726259571979378010504072724396184685687105*10^-30, 
  2.518778851272623621162227285634117940301609670334958*10^-32, 
  2.51877885127262338755865623409266317188334844194942*10^-34, 
  2.5187788512726233641982991289385194366894777279525*10^-36, 
  2.518778851272623361862263418423105080586570333924*10^-38, 
  2.51877885127262336162865984737156364515044562412*10^-40, 
  2.5187788512726233616052994902664095016085871417*10^-42, 
  2.518778851272623361602963454555894087254542116*10^-44, 
  2.51877885127262336160272985098484254582037185*10^-46, 
  2.5187788512726233616027064906277373916892954*10^-48, 
  2.518778851272623361602704154592026876399594*10^-50, 
  2.51877885127262336160270392098845582610468*10^-52, 
  2.5187788512726233616027038976280987334158*10^-54, 
  2.518778851272623361602703895292063147553*10^-56, 
  2.51877885127262336160270389505846082303*10^-58, 
  2.5187788512726233616027038950351129312*10^-60, 
  2.518778851272623361602703895032901548*10^-62}.

This is for the same reason that

NSum[(-1)^n (-1 + 1/n^(10^-x/n) + (10^-x Log[n])/n), {n, 1, Infinity},
   WorkingPrecision -> 70, Method -> "AlternatingSigns"] /. x -> 30

and

NSum[(-1)^n (-1 + n^(10^-x/n) - (10^-x Log[n])/n), {n, 1, Infinity}, 
  WorkingPrecision -> 70, Method -> "AlternatingSigns"] /. x -> 30

give

2.51877885127262336160270389503276304518288666816092953082281396924871\
*10^-62

(For version 3.1 you will have to use x->27 or less.)

(I wonder what closed form starts with 2.518778851272623361602703895?)

While changing the sign of "1:"

NSum[(-1)^n (1 + n^(10^-x/n) - (10^-x Log[n])/n), {n, 1, Infinity}, 
  WorkingPrecision -> 70, Method -> "AlternatingSigns"] /. x -> 30

gives a "divergent sum" of

-1.0000000000000000000000000000000000000000000000000000000000038475945

And another constant, with formulas behaving differently, I noticed is from

Table[NSum[(-1)^n (n^(1/(n + 10^x)) - 1), {n, 1, Infinity}, 
   WorkingPrecision -> 70, 
   Method -> 
    "AlternatingSigns"] - (5 (Log[Pi] - Log[2]))*10^-(x + 1), {x, 1, 
  30}]

which gives

  {-0.002158265971057576296292446726899637434277905244619602754711500291\
  280, -0.00002326878830432348151050186189530613369186033070715442565971\
  5184159, -2.\
  344972498540413669051157226943914337661182682378121265593728211*10^-7,\
   -2.34679530411087431851660597513831629183877508983564770824474081*10^\
  -9, -2.3469777203157053575942148177302233368868838187419725158605417*\
  10^-11, -2.\
  346995963293941260610236486862825114739849987209310140146848*10^-13, \
  -2.34699778760534365532625414499456151604960848632944848056957*10^-15,\
   -2.3469979700366196841185707695593955974492155812208948712378*10^-17,\
   -2.346997988279748644892286166044070017626312946190793075881*10^-19, \
  -2.34699799010406155454860381961973372505412612579214679476*10^-21, \
  -2.3469979902864928456500250473931591201743603058719161031*10^-23, \
  -2.346997990304735974761525064795936837006618410461168616*10^-25, \
  -2.34699799030656028767268864548247023705012450346433171*10^-27, \
  -2.3469979903067427189638051393405861346146650120718647*10^-29, \
  -2.346997990306760962092916790084292349947997733009464*10^-31, \
  -2.34699799030676278640582795517224191773710290844785*10^-33, \
  -2.3469979903067629688371190716811726639785895468696*10^-35, \
  -2.346997990306762987080248183332067096497547977600*10^-37, \
  -2.34699799030676298890456109449715655333023197512*10^-39, \
  -2.3469979903067629890869923856136654991677088242*10^-41, \
  -2.346997990306762989105235514725316393937004272*10^-43, \
  -2.34699799030676298910705982763648148525584608*10^-45, \
  -2.3469979903067629891072422589275980128067172*10^-47, \
  -2.346997990306762989107260502056709849751672*10^-49, \
  -2.34699799030676298910726232636962287534485*10^-51, \
  -2.3469979903067629891072625088009325968910*10^-53, \
  -2.346997990306762989107262527044247758914*10^-55, \
  -2.34699799030676298910726252887042117380*10^-57, \
  -2.3469979903067629891072625290714575021*10^-59, \
  -2.346997990306762989107262529275751003*10^-61}

(Likewise I wonder what closed form starts with 2.34699799030676298910726?)

However here we have a different constant converged to by using the reciprocal

Table[NSum[(-1)^n (1/n^(1/(n + 10^x)) - 1), {n, 1, Infinity}, 
   WorkingPrecision -> 70, 
   Method -> 
    "AlternatingSigns"] + (5 (Log[Pi] - Log[2]))*10^-(x + 1), {x, 1, 
  30}]

giving

{0.0027281794786312486595717428387460304770563641777457765720218456040\
40, 0.0000293550399240151127322722379565607620916546859019350100125647\
09573, 2.9551297193773754366885564592734651504693514196068487319926172\
96*10^-7, 
 2.95707365395153655826155461847169020429215068621347502129971599*10^-\
9, 2.9572678525605426063529814114458282642227732785054465491056943*10^\
-11, 2.957287270464422416597614464460409791592623556524877730647258*\
10^-13, 2.95728921223523164631517763738252605852113445125380329353929*\
10^-15, 2.9572894064121167732298398914267611418621574492035860745636*\
10^-17, 2.957289425829803327952191084785434676290469048265224666794*\
10^-19, 2.95728942777157196384472650970279641869142142361488168100*10^\
-21, 2.9572894279657488272381830467062488252286207798089293916*10^-23,
  2.957289427985166513575570730343167129427371191629855892*10^-25, 
 2.95728942798710828220928991900621614638391196404033743*10^-27, 
 2.9572894279873024590726616420755146114008339317553828*10^-29, 
 2.957289427987321876758998812424474393527194892652000*10^-31, 
 2.95728942798732381852763252943979067109607097234091*10^-33, 
 2.9572894279873240127044959011411265018465393721695*10^-35, 
 2.957289427987324032122182238311258126951706025741*10^-37, 
 2.95728942798732403406395087202827126988436394602*10^-39, 
 2.9572894279873240342581277353999725840002517184*10^-41, 
 2.957289427987324034277545421737142715594072394*10^-43, 
 2.95728942798732403427948719037085973059533356*10^-45, 
 2.9572894279873240342796813672342314505144463*10^-47, 
 2.957289427987324034279700784920568806696226*10^-49, 
 2.95728942798732403427970272668920438421309*10^-51, 
 2.9572894279873240342797029208660863609516*10^-53, 
 2.957289427987324034279702940283958748494*10^-55, 
 2.95728942798732403427970294222758788593*10^-57, 
 2.9572894279873240342797029424403697865*10^-59, 
 2.957289427987324034279702942645837845*10^-61}

I could also ask about the constant 2.957289427987324034279702942.

POSTED BY: Marvin Ray Burns

5 Replies

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Marvin Ray Burns

Marvin Ray Burns, original investigator of the MRB constant

Posted 9 years ago

The last reply is OK as far as Sum[(-1)^n (1/n^(1/(n10^x)) - 1), {n, 1, Infinity}] and Sum[(-1)^n (n^(1/(n10^x)) - 1), {n, 1, Infinity}] are concerned, but what about the MRB constant series, Sum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}] ? Here is what we can say, which is prety much expected: FullSimplify[(f2 = (Sum[(-1)^n (n^(1/n) - 1), {n, 1, 2}])) == (E^(a) /. a -> Log[f2]) == (Sum[(-1)^n (Log[1/f2])^n/n!, {n, 0, Infinity}])] (* True) FullSimplify[(f3 = Sum[(-1)^n (n^(1/n) - 1), {n, 1, 3}]) == (E^(a) /. a -> Log[f3]) == (Sum[(-1)^n (Log[1/f3])^n/n!, {n, 0, Infinity}])] ( True) FullSimplify[(f4 = Sum[(-1)^n (n^(1/n) - 1), {n, 1, 4}]) == (E^(a) /. a -> Log[f4]) == (Sum[(-1)^n (Log[1/f4])^n/n!, {n, 0, Infinity}])] (True) FFullSimplify[{(fI = N[Sum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}]]), fI - Re[E^(a) /. a -> Log[fI]], fI - Re[Sum[(-1)^n (Log[1/fI])^n/n!, {n, 0, Infinity}]], N[fI - Re[ Sum[(-1)^x (Log[x]/x)^n/n!, {x, 1, Infinity}, {n, 1, Infinity}]]]}] ( {0.18786, 0., 0., 3.3306710^-16})

The last reply is OK as far as

Sum[(-1)^n (1/n^(1/(n*10^x)) - 1), {n, 1, Infinity}]

and

Sum[(-1)^n (n^(1/(n*10^x)) - 1), {n, 1, Infinity}]

are concerned, but what about the MRB constant series,

Sum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}]

Here is what we can say, which is prety much expected:

   FullSimplify[(f2 = (Sum[(-1)^n (n^(1/n) - 1), {n, 1,  2}])) 
== (E^(a) /.  a -> Log[f2])
== (Sum[(-1)^n (Log[1/f2])^n/n!, {n, 0, Infinity}])]


 (* True*)

  FullSimplify[(f3 = Sum[(-1)^n (n^(1/n) - 1), {n, 1, 3}]) 
== (E^(a) /.  a -> Log[f3]) 
== (Sum[(-1)^n (Log[1/f3])^n/n!, {n, 0, Infinity}])]

 (* True*)

  FullSimplify[(f4 = Sum[(-1)^n (n^(1/n) - 1), {n, 1, 4}]) 
== (E^(a) /.  a -> Log[f4]) 
== (Sum[(-1)^n (Log[1/f4])^n/n!, {n, 0, Infinity}])]

 (*True*)

  FFullSimplify[{(fI = N[Sum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}]]),
    fI - Re[E^(a) /. a -> Log[fI]],
    fI - Re[Sum[(-1)^n (Log[1/fI])^n/n!, {n, 0, Infinity}]], 
    N[fI - Re[
       Sum[(-1)^x (Log[x]/x)^n/n!, {x, 1, Infinity}, {n, 1, 
         Infinity}]]]}]

(* {0.18786, 0., 0., 3.33067*10^-16}*)

POSTED BY: Marvin Ray Burns