In[]:=
PrimePi[12]
Out[]=
5
In[]:=
12./Log[12]
Out[]=
4.82916
In[]:=
PrimePi[22]
Out[]=
8
In[]:=
22./Log[22]
Out[]=
7.11734
In[]:=
Table[(-1)^n(E^(1/PrimePi[n])-1),{n,2,17}]//TableForm
Out[]//TableForm=
-1+
1-

-1+

1-
1/3

-1+
1/3

1-
1/4

-1+
1/4

1-
1/4

-1+
1/4

1-
1/5

-1+
1/5

1-
1/6

-1+
1/6

1-
1/6

-1+
1/6

1-
1/7

In[]:=
Table[nn=Prime[2n]-1;(-1)^nn(E^(1/PrimePi[nn])-1),{n,2,4}]//TableForm
Out[]//TableForm=
-1+
1/3

-1+
1/5

-1+
1/7

In[]:=
Table[nn=Prime[n];(-1)^nn(E^(Log[nn]/nn)-1),{n,2,4}]//TableForm
Out[]//TableForm=
1-
1/3
3
1-
1/5
5
1-
1/7
7
The
PrimePi
function,whichcountsthenumberofprimenumberslessthanorequaltoagivennumber
n
,isasymptoticallyequivalentto
n/logn
as
n
approachesinfinity.​​​​Thismeansthatforlargevaluesof
n
,theratio
PrimePi(n)
n/logn
approaches1.ThisrelationshipisaresultofthePrimeNumberTheorem,whichdescribesthedistributionofprimenumbersamongtheintegers.
Theseries
N
∑
n2
n
(-1)
(
1/PrimePi[n]
E
-1)
isbeingcomputed,andwe'reobservingthatas
N
increases,theresultalternatesfrom
E-1
to-

+backtoE-1,andalternatetoacloserpointtoE-1.Thenitrepeatsgettingcloser,asshownintheplotbelow.Thisisanalogoustohowtheseries
∞
∑
n1
n
(-1)
(
log(n)/n
E
-1)
convergestotheMRBconstant,exceptheretheconvergenceisto
E-1
duetothesubstitutioninvolvingtheprimecountingfunction.​​
In[]:=
ge=With[{maxN=20},​​ListLinePlot[{Table[E-1,{N,2,maxN}],​​Table[NSum[(-1)^n(E^(1/PrimePi[n])-1),{n,2,N}],{N,2,maxN}]},​​PlotMarkers->Automatic,​​AxesLabel->{"N","Partial Sum"},​​PlotLabel->"ge=Alternating Convergence of the E-1 Series",​​PlotLegends->{"E-1","Partial Sums"}​​]​​]
Out[]=
E-1
Partial Sums
In[]:=
gep=With[{maxN=20},​​ListLinePlot[{Table[E,{N,2,maxN}],​​1+Table[NSum[(-1)^n(E^(1/PrimePi[n])-1),{n,2,N}],{N,2,maxN}]},​​PlotMarkers->Automatic,​​AxesLabel->{"N","Partial Sum"},​​PlotLegends->{"E","E-series Partial Sums"}​​]​​];
In[]:=
m=NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->30]
Out[]=
0.18785964246206712024857897184
In[]:=
gm=With[{maxN=20},​​ListLinePlot[{Table[m,{N,2,maxN}],​​Table[NSum[(-1)^n(E^(Log[n]/n)-1),{n,2,N}],{N,2,maxN}]},​​PlotMarkers->Red,​​AxesLabel->{"N","Partial Sum"},​​PlotLabel->"gm=Alternating Convergence of the MRB constant Series",​​PlotLegends->{{"CMRB","Partial Sums"}}​​]​​]
Out[]=
CMRB
Partial Sums
In[]:=
gmp=With[{maxN=20},​​ListLinePlot[E-m+{Table[m,{N,2,maxN}],​​Table[NSum[(-1)^n(E^(Log[n]/n)-1),{n,2,N}],{N,2,maxN}]},​​PlotMarkers->{Red,Green},​​AxesLabel->{"N","Partial Sum"},​​PlotLegends->{{"CMRB+E-CMRB","CMRB-series Partial Sums +E-CMRB"}}​​]​​];
In[]:=
Show[{gep,gmp},PlotLabel->"ge overlaid with gm"]
Out[]=
E
E-series Partial Sums
CMRB+E-CMRB
CMRB-series Partial Sums +E-CMRB
Since limit of CMRB =CMRB, of course the limit of CMRB-series Partial Sums +E-CMRB=E.
The structural similarities imply that the series share common mathematical features and convergence behaviors, even though they converge to different limits.

Induction

Approach:

1. Alternating Signs:
2. Analyzing the Functions:
3. Recursive Structure:
Let's reason through this in the Wolfram Language to verify:
I will evaluate this expression to see if the inequality holds under these assumptions.
Let's analyze the problem based on the structure of the series:

Proof Outline:

1. Alternating Signs:
2. Asymptotic Behavior:
3. Dominance of Terms:
4. Alternating Series Behavior: