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Somewhat likely looking constructions for the MRB constant
Somewhat likely looking constructions for the MRB constant
NIntegrate-1,{x,1,-InfinityI}//Im
1/x
x
Sin[Pix]
Out[]=
0.18786
NIntegrate-1,{x,1-1/10^2,-InfinityI}//Im
1/x
x
Sin[Pix]
Out[]=
0.18786
NIntegrate-1,{x,1/2,-InfinityI}//Im
1/x
x
Sin[Pix]
Out[]=
0.18786
In[]:=
NIntegrate-1,{x,1/10^2,-InfinityI}//Im
1/x
x
Sin[Pix]
Out[]=
0.18786
NIntegrate-1,{x,1,InfinityI}//Im
1/x
x
Cos[Pi(x+1/2)]
Out[]=
0.18786
NIntegrate-1,{x,1/2,InfinityI}//Im
1/x
x
Cos[Pi(x+1/2)]
Out[]=
0.18786
NSum[Im[(-1)],{x,1,Infinity}]
x+1/2
(-1)
1/x
x
Out[]=
0.18786
NSum-1,{x,1,Infinity}
1/x
x
Sin[Pi(x+1/2)]
Out[]=
0.18786
NSum[Re[(-1)],{x,1,Infinity}]
x
(-1)
1/x
x
Out[]=
0.18786
NSum-1,{x,1,Infinity}
1/x
x
Cos[Pix]
Out[]=
0.18786
In[]:=
Notice two curios of the vertical distances between the partial sums and integrations and their limit-points:
m is approximated by exactly 10 partial sums in special ways. Why 10?
m is approximated by exactly 10 partial sums in special ways. Why 10?
f[x_]=(-1)^xf[x]
Out[]=
x
(-1)
1
x
x
{m,1/2-Sum[f[k],{k,1,10}]}//N
Out[]=
{0.18786,0.186768}
{-m,Sum[Mean[{
f[3k],f[5k],f[7k]
}],{k,1,10}]}//NOut[]=
{-0.18786,-0.183419}
In[]:=
f[x_]=(-1)^x-1;
1
x
x
{m,1/2-Sum[Total[{f[2k],3f[3k],3f[3k],f[4k]}]/11,{k,1,10}]}//N
Out[]=
{0.18786,0.187378}
WolframAlpha["What is the Integral Test?",{{"DefinitionPod:CalculusResult",1},"ComputableData"}]
Out[]=
{Let f(x) be a real-valued function that is continuous and decreasing in [1, infinity) and satisfies f(x)>=0 for all x in [1, infinity). Set a_n = f(n).,1.If integral_1^infinity f(x) dx is convergent, then sum_(n=1)^infinity a_n is convergent.,2.If integral_1^infinity f(x) dx is divergent, then sum_(n=1)^infinity a_n is divergent.,In fact, the test can be stated for any integral [N, infinity), where N is a natural number, instead of [1, infinity).}
Forf(x)=(-1)*Cos[Pi*x]
1/x
x
The integrated analog of m, ∞∫1-1+1xxCos[πx]x, is similarly approximated below.
The integrated analog of m, -1+Cos[πx]x, is similarly approximated below.
∞
∫
1
1
x
x
(Sum[Mean[{f[k],-1)*Cos[Pi*x],{x,1,Infinity},WorkingPrecision10]//RealDigits[#,10,10]&//First
f[3k]
,
f[5k]
,
f[7k]
,f[9k]}],{k,1,10}]-7)//RealDigits[#,10,10]&//First,NIntegrate[(1/x
x
Out[]=
{{7,0,7,7,4,7,7,7,0,0},{7,0,7,7,6,0,3,9,4,0}}
Somewhat likely looking constructions for the MRB constant’s integrated analog
Somewhat likely looking constructions for the MRB constant’s integrated analog
NIntegrate-1,{x,1,InfinityI}
1/x
x
ISin[-Pix]+Cos[-Pix]
Out[]=
0.070776-0.0473806
NIntegrate-1,{x,1,InfinityI}
1/x
x
Exp[-PixI]
Out[]=
0.070776-0.0473806
NIntegrate[Exp[PixI](-1),{x,1,InfinityI}]
1/x
x
Out[]=
0.070776-0.0473806
)
Unlikely looking possible construction for the MRB constant’s integrated analog
Unlikely looking possible construction for the MRB constant’s integrated analog
NIntegrateE^(IPix)-1,{x,1,InfinityI},WorkingPrecision9
1
x
x
Out[]=
0.0707760383-0.0473806179
mkb=NIntegrateE^(IPix)-1,{x,1,InfinityI},WorkingPrecision20
1
x
x
Out[]=
0.070776039311528803669-0.047380617070350786074
What about == the MRB constant analog?
What about == the MRB constant analog?
In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-π];
Table[Print["to ",x," terms gives an error of ",mkb-Sum[N[(I/Pi)^(1-n)*f[n],x],{n,1,x}]],{x,2,12}];
to 2 terms gives an error of 0.001+0.×
-4
10
to 3 terms gives an error of 0.0001+0.×
-5
10
to 4 terms gives an error of 0.×+0.×
-6
10
-6
10
to 5 terms gives an error of 0.×+0.×
-7
10
-7
10