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Mathematics Mathematica Numerical Computation

A few new results involving sums of n^(1/n)

Marvin Ray Burns A.G.S. (cum laude)

Marvin Ray Burns A.G.S. (cum laude), original investigator of the MRB constant

Posted 10 years ago

Reminder, the MRB constant =m=Sum[(-1)^n (n^(1/n)-1)]. In the convergent series of Sum[(-1)^n(n^(x/n) - 1)], Block[{$MaxExtraPrecision = 500}, Table[{NSum[(-1)^n(n^(x/n) - 1), {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 30], N[-3^((x)/3), 30]}, {x, 0, 1000, 10}]] as x gets large the 2 columns go to the same number. Wow!!! According to the algorithm used in NSum by MMA, for any given x (here x=5) in the divergent Sum[(-1)^n(n^(1/n) - xn)] x=5; NSum[(-1)^n(n^(1/n) - xn), {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 30] gives a result the looks like (x - 2)/4 + m. According to the algorithm used in NSum by MMA, in the divergent Sum[(-1)^n(n^(1/(xn)) - n)] Clear[x];Limit[NSum[(-1)^n(n^((1/(n Floor[x]))) - n), {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 60], x -> Infinity] gives a result the looks like - 1/4, while Limit[NSum[(-1)^n(n^((1/(n Floor[x]))) - 1), {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 60], x -> Infinity] gives a result the looks like 0, while for any given base, b>1, (here first 2 then 20), Sum[(-1)^n (n^(1/(nb^x)) - 1), {n, 1, Infinity}] goes to (EulerGammaLog[2] - Log[2]^2/2)b^-x, as x gits large. . Wow! For examples try b = 2; Table[ NSum[(-1)^n (n^(1/(nb^x)) - 1), {n, 1, Infinity}, WorkingPrecision -> 70, Method -> "AlternatingSigns"] - (EulerGammaLog[2] - Log[2]^2/2) b^-x, {x, 1, 20}] and b = 20; Table[ NSum[(-1)^n (n^(1/(nb^x)) - 1), {n, 1, Infinity}, WorkingPrecision -> 70, Method -> "AlternatingSigns"] - (EulerGammaLog[2] - Log[2]^2/2)* b^-x, {x, 1, 20}] Worthy of note are the mantissas of the results of the last two commands; they are similar. (Another wow! here.) For any initial value of cm (here 1) in the iterations of the convergent series, cm=1; l = Table[ cm = NSum[(-1)^n(n^(cm/n) - 1), {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 50], {x, 1, 50}]; l[[50]]/l[[49]] - (EulerGammaLog[2] - Log[2]^2/2) you get a result the looks like 0. For further investigation look at the following beautiful results: In[99]:= b = 10; Table[ NSum[(-1)^n (n^(b^-x/(n)) - n), {n, 1, Infinity}, WorkingPrecision -> 60, Method -> "AlternatingSigns"], {x, 1, 20}] During evaluation of In[99]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(I Interval[{0,2 \[Pi]}]) \[Infinity]. >> During evaluation of In[99]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(I Interval[{0,2 \[Pi]}]) \[Infinity]. >> During evaluation of In[99]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(I Interval[{0,2 \[Pi]}]) \[Infinity]. >> During evaluation of In[99]:= General::stop: Further output of N::meprec will be suppressed during this calculation. >> Out[99]= {-0.23375861648429190757010936707036652336675123442412899836,\ -0.24839878958617333703748478337973121361430612700870214369, \ -0.24984010590587326558560503114709446650140796000537764427, \ -0.24998401285774527616078083446765801163806028451488528910, \ -0.24999840130844379424341261854103342526262694776916459300, \ -0.24999984013107106977791992138913646975004610247981306365, \ -0.24999998401310937387901510144762335410679105478511282376, \ -0.24999999840131096005691142856230269928740293995728161585, \ -0.24999999984013109623238123972773030082495490360018006266, \ -0.24999999998401310962550502493917535503611414714462259528, \ -0.24999999999840131096257317150357924607779399782998430991, \ -0.24999999999984013109625754384045453940084587231537083946, \ -0.24999999999998401310962575665094642008570257673425830549, \ -0.24999999999999840131096257568776365167002412502390932321, \ -0.24999999999999984013109625756900305526361694899403010474, \ -0.24999999999999998401310962575690257242732784039281641775, \ -0.24999999999999999840131096257569027991174244562502315407, \ -0.24999999999999999984013109625756902821786434130916941912, \ -0.24999999999999999998401310962575690282405333522919330385, \ -0.24999999999999999999840131096257569028242800266471178491} In[100]:= b = 10; Table[ NSum[(-1)^n (n^(b^-x/(n)) - x), {n, 1, Infinity}, WorkingPrecision -> 60, Method -> "AlternatingSigns"], {x, 1, 20}] During evaluation of In[100]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -E^(2 I Interval[{0,\[Pi]}]). >> During evaluation of In[100]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -2 E^(2 I Interval[{0,\[Pi]}]). >> During evaluation of In[100]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -3 E^(2 I Interval[{0,\[Pi]}]). >> During evaluation of In[100]:= General::stop: Further output of N::meprec will be suppressed during this calculation. >> Out[100]= \ {0.01624138351570809242989063292963347663324876572268322487416, \ 0.501601210413826662962515216620268786385693873138092735083, \ 1.000159894094126734414394968852905533498592040141399890032, \ 1.500015987142254723839219165532341988361939715631874900741, \ 2.000001598691556205756587381458966574737373052377578252379, \ 2.500000159868928930222080078610863530249953897666912437257, \ 3.000000015986890626120984898552376645893208945361595332684, \ 3.500000001598689039943088571437697300712597060189409196134, \ 4.000000000159868903767618760272269699175045096546493404855, \ 4.50000000001598689037449497506082464496388585300203352777, \ 5.00000000000159868903742682849642075392220600231665446868, \ 5.50000000000015986890374245615954546059915412783125059466, \ 6.00000000000001598689037424334905357991429742341234578416, \ 6.50000000000000159868903742431223634832997587512267742198, \ 7.00000000000000015986890374243099694473638305115253929599, \ 7.50000000000000001598689037424309742757267215975373563851, \ 8.00000000000000000159868903742430972008825755452151155772, \ 8.50000000000000000015986890374243097178213565883734794821, \ 9.00000000000000000001598689037424309717594666491730671902, \ 9.50000000000000000000159868903742430971757199748177089350} In[96]:= b = 10; Table[ NSum[(-1)^n (n^(b^-x/(n)) - b x), {n, 1, Infinity}, WorkingPrecision -> 60, Method -> "AlternatingSigns"], {x, 1, 20}] During evaluation of In[96]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -9 E^(2 I Interval[{0,\[Pi]}]). >> During evaluation of In[96]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -19 E^(2 I Interval[{0,\[Pi]}]). >> During evaluation of In[96]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -29 E^(2 I Interval[{0,\[Pi]}]). >> During evaluation of In[96]:= General::stop: Further output of N::meprec will be suppressed during this calculation. >> Out[96]= {4.51624138351570809242989063292963347663324876572252712469, \ 9.50160121041382666296251521662026878638569387313778053471, \ 14.50015989409412673441439496885290553349859204014093158948, \ 19.50001598714225472383921916553234198836193971563125050000, \ 24.50000159869155620575658738145896657473737305237679775146, \ 29.50000015986892893022208007861086353024995389766597583615, \ 34.50000001598689062612098489855237664589320894536050263139, \ 39.50000000159868903994308857143769730071259706018816039466, \ 44.5000000001598689037676187602722696991750450965450885032, \ 49.5000000000159868903744949750608246449638858530004725259, \ 54.5000000000015986890374268284964207539222060023149373667, \ 59.5000000000001598689037424561595454605991541278293773924, \ 64.5000000000000159868903742433490535799142974234103164818, \ 69.5000000000000015986890374243122363483299758751204920194, \ 74.5000000000000001598689037424309969447363830511501977932, \ 79.5000000000000000159868903742430974275726721597512380356, \ 84.5000000000000000015986890374243097200882575545188578546, \ 89.5000000000000000001598689037424309717821356588345381449, \ 94.5000000000000000000159868903742430971759466649143408155, \ 99.5000000000000000000015986890374243097175719974786488898} And last but not least, In[101]:= b = 10; Table[ NSum[(-1)^n (n^(b^-x/(n)) - n x), {n, 1, Infinity}, WorkingPrecision -> 60, Method -> "AlternatingSigns"], {x, 1, 20}] During evaluation of In[101]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(I Interval[{0,2 \[Pi]}]) \[Infinity]. >> During evaluation of In[101]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(2 I Interval[{0,\[Pi]}]) (-\[Infinity]). >> During evaluation of In[101]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(I Interval[{0,2 \[Pi]}]) (-\[Infinity]). >> During evaluation of In[101]:= General::stop: Further output of N::meprec will be suppressed during this calculation. >> Out[101]= \ {-0.23375861648429190757010936707036652336675123442412899836, \ 0.0016012104138266629625152166202687863856938728444682886, \ 0.2501598940941267344143949688529055334985920397009632203, \ 0.5000159871422547238392191655323419883619397150446260078, \ 0.7500015986915562057565873814589665747373730516435171362, \ 1.0000001598689289302220800786108635302499538967860390979, \ 1.2500000159868906261209848985523766458932089443339097700, \ 1.5000000015986890399430885714376973007125970590149114103, \ 1.7500000001598689037676187602722696991750450952251833957, \ 2.0000000000159868903744949750608246449638858515339112954, \ 2.2500000000015986890374268284964207539222060007017200131, \ 2.5000000000001598689037424561595454605991541260695039158, \ 2.7500000000000159868903742433490535799142974215037868821, \ 3.0000000000000015986890374243122363483299758730673062967, \ 3.2500000000000001598689037424309969447363830489503559475, \ 3.5000000000000000159868903742430974275726721574047400668, \ 3.7500000000000000015986890374243097200882575520257037627, \ 4.000000000000000000159868903742430971782135656194727930, \ 4.250000000000000000015986890374243097175946662127874478, \ 4.500000000000000000001598689037424309717571994545526429} Notice the 15986890374243097... in the above; EulerGammaLog[2] - Log[2]^2/2== 0.15986890374243097176.... Also notice the 84013109625756902824...; 1 - (EulerGammaLog[2] - Log[2]^2/2)=0.84013109625756902824....

Reminder, the MRB constant =m=Sum[(-1)^n (n^(1/n)-1)].

In the convergent series of Sum[(-1)^n*(n^(x/n) - 1)],

Block[{$MaxExtraPrecision = 500}, 
Table[{NSum[(-1)^n*(n^(x/n) - 1), {n, 1, Infinity}, 
Method -> "AlternatingSigns", WorkingPrecision -> 30], 
N[-3^((x)/3), 30]}, {x, 0, 1000, 10}]]

as x gets large the 2 columns go to the same number. Wow!!!

According to the algorithm used in NSum by MMA, for any given x (here x=5) in the divergent Sum[(-1)^n(n^(1/n) - xn)]

x=5; NSum[(-1)^n*(n^(1/n) - x*n), {n, 1, Infinity}, 
Method -> "AlternatingSigns", WorkingPrecision -> 30]

gives a result the looks like (x - 2)/4 + m.

According to the algorithm used in NSum by MMA, in the divergent Sum[(-1)^n(n^(1/(xn)) - n)]

Clear[x];Limit[NSum[(-1)^n*(n^((1/(n Floor[x]))) - n), {n, 1, Infinity}, 
  Method -> "AlternatingSigns", WorkingPrecision -> 60], 
 x -> Infinity]

gives a result the looks like - 1/4, while

Limit[NSum[(-1)^n*(n^((1/(n Floor[x]))) - 1), {n, 1, Infinity}, 
  Method -> "AlternatingSigns", WorkingPrecision -> 60], 
 x -> Infinity]

gives a result the looks like 0, while for any given base, b>1, (here first 2 then 20), Sum[(-1)^n (n^(1/(nb^x)) - 1), {n, 1, Infinity}] goes to (EulerGammaLog[2] - Log[2]^2/2)*b^-x, as x gits large. . Wow!

For examples try

b = 2; Table[
 NSum[(-1)^n (n^(1/(n*b^x)) - 1), {n, 1, Infinity}, 
   WorkingPrecision -> 70, 
   Method -> "AlternatingSigns"] - (EulerGamma*Log[2] - Log[2]^2/2)*
   b^-x, {x, 1, 20}]

and

b = 20; Table[
 NSum[(-1)^n (n^(1/(n*b^x)) - 1), {n, 1, Infinity}, 
   WorkingPrecision -> 70, 
   Method -> "AlternatingSigns"] - (EulerGamma*Log[2] - Log[2]^2/2)*
   b^-x, {x, 1, 20}]

Worthy of note are the mantissas of the results of the last two commands; they are similar. (Another wow! here.)

For any initial value of cm (here 1) in the iterations of the convergent series,

  cm=1; l = Table[
  cm = NSum[(-1)^n*(n^(cm/n) - 1), {n, 1, Infinity}, 
    Method -> "AlternatingSigns", WorkingPrecision -> 50], {x, 1, 
   50}]; l[[50]]/l[[49]] - (EulerGamma*Log[2] - Log[2]^2/2)

you get a result the looks like 0.

For further investigation look at the following beautiful results:

In[99]:= b = 10; Table[
 NSum[(-1)^n (n^(b^-x/(n)) - n), {n, 1, Infinity}, 
  WorkingPrecision -> 60, Method -> "AlternatingSigns"], {x, 1, 20}]

During evaluation of In[99]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(I Interval[{0,2 \[Pi]}]) \[Infinity]. >>

During evaluation of In[99]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(I Interval[{0,2 \[Pi]}]) \[Infinity]. >>

During evaluation of In[99]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(I Interval[{0,2 \[Pi]}]) \[Infinity]. >>

During evaluation of In[99]:= General::stop: Further output of N::meprec will be suppressed during this calculation. >>

Out[99]= {-0.23375861648429190757010936707036652336675123442412899836,\
 -0.24839878958617333703748478337973121361430612700870214369, \
-0.24984010590587326558560503114709446650140796000537764427, \
-0.24998401285774527616078083446765801163806028451488528910, \
-0.24999840130844379424341261854103342526262694776916459300, \
-0.24999984013107106977791992138913646975004610247981306365, \
-0.24999998401310937387901510144762335410679105478511282376, \
-0.24999999840131096005691142856230269928740293995728161585, \
-0.24999999984013109623238123972773030082495490360018006266, \
-0.24999999998401310962550502493917535503611414714462259528, \
-0.24999999999840131096257317150357924607779399782998430991, \
-0.24999999999984013109625754384045453940084587231537083946, \
-0.24999999999998401310962575665094642008570257673425830549, \
-0.24999999999999840131096257568776365167002412502390932321, \
-0.24999999999999984013109625756900305526361694899403010474, \
-0.24999999999999998401310962575690257242732784039281641775, \
-0.24999999999999999840131096257569027991174244562502315407, \
-0.24999999999999999984013109625756902821786434130916941912, \
-0.24999999999999999998401310962575690282405333522919330385, \
-0.24999999999999999999840131096257569028242800266471178491}

In[100]:= b = 10; Table[
 NSum[(-1)^n (n^(b^-x/(n)) - x), {n, 1, Infinity}, 
  WorkingPrecision -> 60, Method -> "AlternatingSigns"], {x, 1, 20}]

During evaluation of In[100]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -E^(2 I Interval[{0,\[Pi]}]). >>

During evaluation of In[100]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -2 E^(2 I Interval[{0,\[Pi]}]). >>

During evaluation of In[100]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -3 E^(2 I Interval[{0,\[Pi]}]). >>

During evaluation of In[100]:= General::stop: Further output of N::meprec will be suppressed during this calculation. >>

Out[100]= \
{0.01624138351570809242989063292963347663324876572268322487416, \
0.501601210413826662962515216620268786385693873138092735083, \
1.000159894094126734414394968852905533498592040141399890032, \
1.500015987142254723839219165532341988361939715631874900741, \
2.000001598691556205756587381458966574737373052377578252379, \
2.500000159868928930222080078610863530249953897666912437257, \
3.000000015986890626120984898552376645893208945361595332684, \
3.500000001598689039943088571437697300712597060189409196134, \
4.000000000159868903767618760272269699175045096546493404855, \
4.50000000001598689037449497506082464496388585300203352777, \
5.00000000000159868903742682849642075392220600231665446868, \
5.50000000000015986890374245615954546059915412783125059466, \
6.00000000000001598689037424334905357991429742341234578416, \
6.50000000000000159868903742431223634832997587512267742198, \
7.00000000000000015986890374243099694473638305115253929599, \
7.50000000000000001598689037424309742757267215975373563851, \
8.00000000000000000159868903742430972008825755452151155772, \
8.50000000000000000015986890374243097178213565883734794821, \
9.00000000000000000001598689037424309717594666491730671902, \
9.50000000000000000000159868903742430971757199748177089350}

In[96]:= b = 10; Table[
 NSum[(-1)^n (n^(b^-x/(n)) - b x), {n, 1, Infinity}, 
  WorkingPrecision -> 60, Method -> "AlternatingSigns"], {x, 1, 20}]

During evaluation of In[96]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -9 E^(2 I Interval[{0,\[Pi]}]). >>

During evaluation of In[96]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -19 E^(2 I Interval[{0,\[Pi]}]). >>

During evaluation of In[96]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -29 E^(2 I Interval[{0,\[Pi]}]). >>

During evaluation of In[96]:= General::stop: Further output of N::meprec will be suppressed during this calculation. >>

Out[96]= {4.51624138351570809242989063292963347663324876572252712469, \
9.50160121041382666296251521662026878638569387313778053471, \
14.50015989409412673441439496885290553349859204014093158948, \
19.50001598714225472383921916553234198836193971563125050000, \
24.50000159869155620575658738145896657473737305237679775146, \
29.50000015986892893022208007861086353024995389766597583615, \
34.50000001598689062612098489855237664589320894536050263139, \
39.50000000159868903994308857143769730071259706018816039466, \
44.5000000001598689037676187602722696991750450965450885032, \
49.5000000000159868903744949750608246449638858530004725259, \
54.5000000000015986890374268284964207539222060023149373667, \
59.5000000000001598689037424561595454605991541278293773924, \
64.5000000000000159868903742433490535799142974234103164818, \
69.5000000000000015986890374243122363483299758751204920194, \
74.5000000000000001598689037424309969447363830511501977932, \
79.5000000000000000159868903742430974275726721597512380356, \
84.5000000000000000015986890374243097200882575545188578546, \
89.5000000000000000001598689037424309717821356588345381449, \
94.5000000000000000000159868903742430971759466649143408155, \
99.5000000000000000000015986890374243097175719974786488898}

And last but not least,

In[101]:= b = 10; Table[
 NSum[(-1)^n (n^(b^-x/(n)) - n x), {n, 1, Infinity}, 
  WorkingPrecision -> 60, Method -> "AlternatingSigns"], {x, 1, 20}]

During evaluation of In[101]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(I Interval[{0,2 \[Pi]}]) \[Infinity]. >>

During evaluation of In[101]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(2 I Interval[{0,\[Pi]}]) (-\[Infinity]). >>

During evaluation of In[101]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating E^(I Interval[{0,2 \[Pi]}]) (-\[Infinity]). >>

During evaluation of In[101]:= General::stop: Further output of N::meprec will be suppressed during this calculation. >>

Out[101]= \
{-0.23375861648429190757010936707036652336675123442412899836, \
0.0016012104138266629625152166202687863856938728444682886, \
0.2501598940941267344143949688529055334985920397009632203, \
0.5000159871422547238392191655323419883619397150446260078, \
0.7500015986915562057565873814589665747373730516435171362, \
1.0000001598689289302220800786108635302499538967860390979, \
1.2500000159868906261209848985523766458932089443339097700, \
1.5000000015986890399430885714376973007125970590149114103, \
1.7500000001598689037676187602722696991750450952251833957, \
2.0000000000159868903744949750608246449638858515339112954, \
2.2500000000015986890374268284964207539222060007017200131, \
2.5000000000001598689037424561595454605991541260695039158, \
2.7500000000000159868903742433490535799142974215037868821, \
3.0000000000000015986890374243122363483299758730673062967, \
3.2500000000000001598689037424309969447363830489503559475, \
3.5000000000000000159868903742430974275726721574047400668, \
3.7500000000000000015986890374243097200882575520257037627, \
4.000000000000000000159868903742430971782135656194727930, \
4.250000000000000000015986890374243097175946662127874478, \
4.500000000000000000001598689037424309717571994545526429}

Notice the 15986890374243097... in the above; EulerGammaLog[2] - Log[2]^2/2== 0.15986890374243097176.... Also notice the 84013109625756902824...; 1 - (EulerGammaLog[2] - Log[2]^2/2)=0.84013109625756902824....

POSTED BY: Marvin Ray Burns A.G.S. (cum laude)

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