ExploringCrandall'sFirstFormulafortheMRBConstant​​CMRB=S=
∑
m
>=
1
m
(-1)
m!
(m)
η
(m):​​SplittingtheSeriesandIntegralRepresentations

AccordingtotheWolframDocumentation:
Asymptotic
istypicallyusedtosolveproblemsforwhichnoexactsolutioncanbefoundortogetsimpleranswersforcomputation,comparisonandinterpretation.Insuchcases,anasymptoticapproximationoftengivesenoughinformationforsimplifyingorsolvingapplicationproblems.
Let m be the MRB constant, the sum of
1/k
k
-1
exp(πk)
.
SInce a exact solution for the MRB constant in terms of elementary functions seems illusive,
In[]:=
S=
∞
∑
k=1
1/k
k
-1
exp(πk)
Out[]=
∞
∑
k=1
-kπ

-1+
1
k
k

We have the following nontrivial relations:
Asymptotic[
1/k
k
-1,k∞]==
log(k)
k
(theAsymptoticterm)
In[]:=
Asymptotic[
1/k
k
-1,k∞]==
log(k)
k
Out[]=
True
In[]:=
eta1=Sum
log(k)
kexp(πk)
,{k,1,Infinity}
Out[]=
EulerGammaLog[2]-
2
Log[2]
2
Surprisingly, of which, means
∞
∑
k=1
Asymptotic
1/k
k
-1
exp(πk)
,k∞​​+
∞
∑
k=1
1/k
k
-
log(k)
k
-1
exp(πk)
​​=
∞
∑
k=1
1/k
k
-1
exp(πk)
Inessence,thisequalityshowsthattheoriginalseriescanbesplitintoanasymptoticcomponentof
1/k
k
-1
exp(πk)
==eta1,andaremainder
∞
∑
k=1
1/k
k
-
log(k)
k
-1
exp(πk)
==
∑
m
>
1
m
(-1)
m!
(m)
η
(m),andthesecomponentstogetherreconstructtheoriginalseries,convergingtotheMRBconstant,​​​​
∑
m
>=
1
m
(-1)
m!
(m)
η
(m)=S.
shown next
In[]:=
Neta1+
∞
∑
k=1
1/k
k
-
Log[k]
k
-1
Exp[πk]
==S//Chop
Out[]=
True

Concerning an analog of the above sum

ThefollowingadditioncodeevaluatsafunctionthatinvolvestheMeijerGfunctionandcomparingittoanumericalintegralanalogoftheMRBconstant(theanalog).Let'sbreakdownwhat'shappens:​​​​1.FunctionDefinition:​​
• The function
f[n]
is defined using the
MeijerG
function. The Meijer G-function is a very general function that includes many special functions as special cases. It is used here in a complex configuration.
​​
• The function is then used in a sum that depends on a complex power of
I
(the imaginary unit) to equal the analog.
​​2.TableEvaluation:​​
• Two tables are computed. Each table evaluates the difference between a sum involving
f[n]
and a numerical integral.
​​
• The first table evaluates the sum from
n = 1
to x and compares it to an integral with the integrand
(-1)^n (n^(1/n)-1)
.
​​
•
The second table evaluates the sum from
n = 2
to
x
and compares it to an integral with the integrand
(-1)^n (n^(1/n)-(the Asymptotic termused in the MRB constant sum at the beginning of this notebook.)
.
​​3.ComplexResults:​​
• The results of the table evaluations are complex numbers, showing both real and imaginary parts. The use of
N
ensures these are numerical approximations with a specified precision.
​​4.MagnitudeCalculation:​​
• Following each table, the
Abs
function is applied to the results to obtain the magnitudes of the complex numbers. This gives a sense of the size of the differences.
​
​
MeijerG
[{{
a
1
,…,
a
n
},{
a
n+1
,…,
a
p
}},{{
b
1
,…,
b
m
},{
b
m+1
,…,
b
q
}},z]
​is the Meijer G-function
m,n
G
p,q
z
a
1
,…,
a
p
b
1
,…,
b
q
.
M1
In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-π];Table[N[Sum[(I/Pi)^(1-n)*f[n],{n,1,x}],x+6]-NIntegrate[
n
(-1)
(
1/n
n
-1),{n,1,xI},WorkingPrecision->x+6],{x,1,11}]
Out[]=
{-0.0060083+0.0352107,-0.00100558+3.55×
-6
10
,-0.000093587-0.000070483,-5.7199×
-6
10
-7.6050×
-6
10
,-2.6165×
-7
10
-5.5902×
-7
10
,-8.972×
-9
10
-3.2745×
-8
10
,-1.988×
-10
10
-1.6159×
-9
10
,6.8×
-13
10
-6.888×
-11
10
,3.07×
-13
10
-2.651×
-12
10
,2.75×
-14
10
-9.89×
-14
10
,1.29×
-15
10
-2.50×
-15
10
}
In[]:=
Abs[%]
Out[]=
{0.0357197,0.00100559,0.000117160,9.5159×
-6
10
,6.1722×
-7
10
,3.3952×
-8
10
,1.6281×
-9
10
,6.889×
-11
10
,2.669×
-12
10
,1.027×
-13
10
,2.81×
-15
10
}
M2