CMRB=(-1)
∞
∑
x=1
x
(-1)
1/x
x
CMRB=-1k!-kΓ(k)-Γk,
lim
k∞
∞
∑
x=1
x
(-1)
1
x
x
1
x
x
log(x)
x
k(k-1)!
CMRB=(-1)Γk,
lim
k∞
∞
∑
x=1
x
(-1)
1/x
x
log(x)
x
Γ(k)
It seems that for optimal performance in computing d digits, use the following; later listed is better.
CMRB=-1k!-kΓ(k)-Γk,/.k.
∞
∑
x=1
x
(-1)
1
x
x
1
x
x
log(x)
x
k(k-1)!
d
2
CMRB=(-1)Γk,/.k2d
∞
∑
x=1
x
(-1)
1/x
x
log(x)
x
Γ(k)
CMRB=(-1)Γk,/.k
∞
∑
x=1
x
(-1)
1/x
x
log(x)
x
Γ(k)
7d
8
CMRB=--1log(x)+-1/.k
∞
∑
x=1
x
(-1)
1
x
x
Γ(k)
1/x
x
d
2
In[]:=
ClearSystemCache[]
In[]:=
f[x_]=(-1)^x(x^(1/x)-1);
In[]:=
Timing[CMRB=NSum[f[x],{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision2000];]
Out[]=
{0.640625,Null}
In[]:=
Timing[CMRB-NSum[(((-1)^x*((-1+x^x^(-1))*k!-k*x^x^(-1)*(Gamma[k]-Gamma[k,Log[x]/x])))/(k*(-1+k)!))/.k500,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision1000]]
Out[]=
{1.8125,-3.×}
-998
10
In[]:=
Timing[CMRB-NSum[(((-1)^x*((-1+x^x^(-1))*k!-k*x^x^(-1)*(Gamma[k]-Gamma[k,Log[x]/x])))/(k*(-1+k)!))/.k1000,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision2000]]
Out[]=
{7.5625,0.×}
-1999
10
In[]:=
Assuming[k∈Integers&&k>0,FullSimplify[(((-1)^x*((-1+x^x^(-1))*k!-k*x^x^(-1)*(Gamma[k]-Gamma[k,Log[x]/x])))/(k*(-1+k)!))]]
Out[]=
x
(-1)
1
x
x
Log[x]
x
Gamma[k]
CMRB=-1+Gammak,/.k2d
∞
∑
x=1
x
(-1)
1
x
x
Log[x]
x
Gamma[k]
In[]:=
ClearSystemCache[]
In[]:=
f[x_]=(-1)^x(x^(1/x)-1);
In[]:=
Timing[CMRB=NSum[f[x],{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision1000];]
Out[]=
{0.6875,Null}
In[]:=
ClearSystemCache[]
In[]:=
Timing[CMRB-NSum[((f[x]*Gamma[k,Log[x]/x])/Gamma[k])/.k2000,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision1000]]
Out[]=
{1.73438,0.×}
-999
10
In[]:=
ClearSystemCache[]
In[]:=
Timing[CMRB-NSum[((f[x]*Gamma[k,Log[x]/x])/Gamma[k])/.k4000,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision2000]]
Out[]=
{7.04688,3.×}
-998
10
CMRB=-1+Gammak,/.kd
∞
∑
x=1
x
(-1)
1
x
x
Log[x]
x
Gamma[k]
7
8
ClearSystemCache[]
In[]:=
Timing[CMRB-NSum[((f[x]*Gamma[k,Log[x]/x])/Gamma[k])/.k3500,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision2000]]
Out[]=
{6.21875,0.×}
-1999
10
CMRB=--1log(x)+-1/.k
∞
∑
x=1
x
(-1)
1
x
x
Γ(k)
1/x
x
d
2
In[]:=
Timing[CMRB-NSum[(-1)^x*(-1+x^x^(-1)-(x^(-1+x^(-1))*Log[x])/Gamma[k])/.k500,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision1000]]
Out[]=
{0.984375,0.×}
-999
10