Relative Convergence Rate
Relative Convergence Rate
In[]:=
DecimalapproximationoftheMRBconstant
Out[]=
approximationconstantDecimalMRBofthe
In[]:=
m=WolframAlpha["MRB constant",{{"DecimalApproximation",1},"ComputableData"}]
Out[]=
0.1878596424620671202485179340542732300559030949001387861720046840894772315646602
In[]:=
Precision[m]
Out[]=
79.
Error using 10^n terms
Error using 10^n terms
In[]:=
p=Table[m-NSum[(-1),{n,10^x},WorkingPrecision200,PrecisionGoal200,Method"AlternatingSigns"],{x,6,33}]//N//Short
n
(-1)
1/n
n
Out[]//Short=
{-6.9078×,-8.05905×,-9.21034×,23,-3.68414×,-3.79927×}
-6
10
-7
10
-8
10
-31
10
-32
10
In[]:=
c1=ActivateLimitDerivative[n][DirichletEta][x]/.n1,x1,Unevaluated[Sum]
n
(-1)
n!
Out[]=
1
2
In[]:=
f1[n_]:=Derivative[n][DirichletEta][n];
n
(-1)
n!
Error using just n terms of the first Crandall formula
Error using just n terms of the first Crandall formula
In[]:=
(q=Table[m+(c1+Sum[f1[n],{n,2,x}]),{x,6,33}]//N)//Short
Out[]//Short=
{1.87833×,4.54955×,24,-3.21255×,-3.5388×}
-8
10
-10
10
-52
10
-54
10
An average ratio of the largest partial sums by each method
An average ratio of the largest partial sums by each method
In[]:=
me=MantissaExponent;
In[]:=
s0=Table[me[q[[-n]]][[2]]/me[p[[-n]]][[2]]//N,{n,2,26}]//Mean
Out[]=
1.61795
In[]:=
GoldenRatio//N
Out[]=
1.61803
In[]:=
t1="For partial sums ranging from 10^6 to 10^33 terms, Crandall's formula converges ";t2="approximately a factor of the golden ratio faster per decade increase in the number of terms than direct summation.";Print[Style[t1,Red],Style[GoldenRatio/s0//PercentForm,Large],Style[t2,Blue]]
For partial sums ranging from 10^6 to 10^33 terms, Crandall's formula converges 100%approximately a factor of the golden ratio faster per decade increase in the number of terms than direct summation.
In[]:=