In[]:=
f[x_]=(-1)^x(x^(1/x)-1);
In[]:=
CMRB=NSum[f[x],{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision100]
Out[]=
0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443217278
In[]:=
l=TableSum(-1)^xLog[x]Sum[k!/(n)!x^(k-1-n)Log[x]^(n-1)/k,{n,1,k-1}],{x,1,Infinity},{k,1,10}//Short//TableForm
1
(k-1)!
k-1
x
Out[]//TableForm=
0,(2EulerGammaLog[2]-),6,,
1
2
2
Log[2]
1
4800
82+1
48771072000
In[]:=
(la=Expand[Table[l[[n+1]]-l[[n]],{n,1,9}]])//TableForm
Out[]//TableForm=
EulerGammaLog[2]- 2 Log[2] 2 |
- 1 12 2 π 1 24 2 π 2 Log[2] 2 π 1 12 2 π ′′ Zeta 4 |
1 24 3 Log[2] 1 8 2 Log[2] ′ Zeta 1 24 ′′ Zeta 1 8 (3) Zeta |
4 π 4 Log[2] 17280 1 48 3 Log[2] ′ Zeta 1 32 2 Log[2] ′′ Zeta 1 48 (3) Zeta 7 192 (4) Zeta |
5 Log[2] 1920 1 384 4 Log[2] ′ Zeta 1 192 3 Log[2] ′′ Zeta 1 192 2 Log[2] (3) Zeta Log[32] (4) Zeta 1920 1 128 (5) Zeta |
6 π 6 Log[2] 21772800 5 Log[2] ′ Zeta 3840 4 Log[2] ′′ Zeta 1536 3 Log[2] (3) Zeta 1152 2 Log[2] (4) Zeta 1536 Log[2] (5) Zeta 3840 31 (6) Zeta 23040 |
7 Log[2] 322560 6 Log[2] ′ Zeta 46080 5 Log[2] ′′ Zeta 15360 4 Log[2] (3) Zeta 9216 3 Log[2] (4) Zeta 9216 2 Log[2] (5) Zeta 15360 Log[2] (6) Zeta 46080 (7) Zeta 5120 |
8 π 8 Log[2] 48771072000 7 Log[2] ′ Zeta 645120 6 Log[2] ′′ Zeta 184320 5 Log[2] (3) Zeta 92160 4 Log[2] (4) Zeta 73728 3 Log[2] (5) Zeta 92160 2 Log[2] (6) Zeta 184320 Log[2] (7) Zeta 645120 127 (8) Zeta 5160960 |
9 Log[2] 92897280 8 Log[2] ′ Zeta 10321920 7 Log[2] ′′ Zeta 2580480 6 Log[2] (3) Zeta 1105920 5 Log[2] (4) Zeta 737280 4 Log[2] (5) Zeta 737280 3 Log[2] (6) Zeta 1105920 2 Log[2] (7) Zeta 2580480 Log[2] (8) Zeta 10321920 17 (9) Zeta 6193152 |
In[]:=
(ll=Table[(Sum[(-1)^xLog[x]^n/(n!x^n),{x,1,Infinity}]),{n,1,9}])//Short//TableForm
Out[]//TableForm=
(2EulerGammaLog[2]-),(-1-61),6,
1
2
2
Log[2]
1
24
2
π
2
1
1
92897280
In[]:=
Expand[Table[ll[[n]]-l[[n+1]]+l[[n]],{n,1,9}]]
Out[]=
{0,0,0,0,0,0,0,0,0}
In[]:=
Expand[Table[ll[[n]]-la[[n]],{n,1,9}]]
Out[]=
{0,0,0,0,0,0,0,0,0}
In[]:=
CMRB-(l)//N
Out[]=
{0.18786,0.0279907,0.00280295,0.000207355,0.0000118838,5.3695×,1.87833×,4.54955×,2.79824×,-4.4772×}
-7
10
-8
10
-10
10
-12
10
-13
10
In[]:=
(ll)//N
Out[]=
{0.159869,0.0251878,0.0025956,0.000195471,0.0000113469,5.18166×,1.83284×,4.52157×,3.24596×}
-7
10
-8
10
-10
10
-12
10
In[]:=
N[CMRB-(ll+l[[1;;9]]),30]
Out[]=
{0.0279907387196361484915700637294,0.00280295020690991487554302477903,0.000207354973003898926467754641480,0.0000118838220054139752268887360275,5.36949668548612943348287695556×,1.87833475995791223751662833641×,4.54955477824640390711570422552×,2.79823510102935621930983118607×,-4.47720121506293658470300505848×}
-7
10
-8
10
-10
10
-12
10
-13
10
In[]:=
Chop[%]
Out[]=
{0.0279907387196361484915700637294,0.00280295020690991487554302477903,0.000207354973003898926467754641480,0.0000118838220054139752268887360275,5.36949668548612943348287695556×,1.87833475995791223751662833641×,4.54955477824640390711570422552×,0,0}
-7
10
-8
10
-10
10