TheMRBconstantanalogtotheExponentialfunction
The MRB constant (sum of odd and even roots). m=
The MRB constant (sum of odd and even roots). m=
In[]:=
m=(-1)
∞
∑
n=1
n
(-1)
1/n
n
;
In[]:=
(N[m]-N[Sum[(2*n)^(1/(2*n))-(2*n-1)^(1/(2*n-1)),{n,1,Infinity}],14])//Chop
Out[]=
0
Exp[x]: (sum of odd and even powers). x=
Exp[x]: (sum of odd and even powers). =
x
In[]:=
TrigToExp[Activate[Inactive[Sum][x^(2*n.)/(2*n.)!,{n.,0,Infinity}]+Inactive[Sum][x^(2*n.+1)/(2*n.+1)!,{n.,0,Infinity}],Unevaluated[Sum]]]
Out[]=
x
a=Asymptotic1xx-1,{x,Infinity,1}. A Relationship among the two parts of the Analog.b= m=.
a=
Asymptotic-1,{x,Infinity,1}
1
x
x
In[]:=
a=Asymptotic[x^(1/x)-1,{x,Infinity,1}];
In[]:=
b=Activate[(-(1/2)+Inactive[Sum][x^(2*n.)/(2*n.)!,{n.,0,Infinity}]/.x->a)+(-(1/2)+Inactive[Sum][x^(2*n.+1)/(2*n.+1)!,{n.,0,Infinity}]/.x->a),Unevaluated[Sum]];
In[]:=
N[m==Sum[(-1)^x*b,{x,1,Infinity}]]
Out[]=
True