Flint Hills Series

Author

Eric W. Weisstein
December 6, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/FlintHillsSeries.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FlintHillsSeries.html.

©2022 Wolfram Research, Inc. except for portions noted otherwise

Sources

https://www.researchgate.net/publication/366016235_Proof_of_the_Convergence_of_the_Flint-Hills_Series%27The_Irrationality_Measure_of_p_is_less_than_or_equal_to_25%27?channel=doi&linkId=638e02bd11e9f00cda1f2471&showFulltext=true

claims

N[Zeta[5]+(π/9)-(2/(45π)),20]

?!

Numeric value

NSum[(1/((n^3)((Sin[n])^2))),{n,∞}]

NSum[(1/((n^3)((Sin[n])^2))),{n,∞},WorkingPrecision->20]

Total[N[Table[(1/((n^3)((Sin[n])^2))),{n,100}],20]]//Timing

Total[N[Table[(1/((n^3)((Sin[n])^2))),{n,10^4}],20]]//Timing

Total[N[Table[(1/((n^3)((Sin[n])^2))),{n,10^5}],20]]//Timing

Total[N[Table[(1/((n^3)((Sin[n])^2))),{n,10^6}],20]]//Timing

Total[N[Table[(1/((n^3)((Sin[n])^2))),{n,10^7}],20]]//Timing

Total[N[Table[(1/((n^3)((Sin[n])^2))),{n,10^8}],20]]//Timing

(∑_(n=1)^∞)(1/((n^3) (sin^2)(n)))

Sum

V5

Sum[(1/((n^3)((Sin[n])^2))),{n,∞}]

V7

Sum[(1/((n^3)((Sin[n])^2))),{n,∞}]//Timing

...wrong (unknown)...

SumConvergence[(1/((n^3)((Sin[n])^2))),n]//Timing

...wrong (unknown)...

V8

Sum[(1/((n^3)((Sin[n])^2))),{n,∞}]//Timing

SumConvergence[(1/((n^3)((Sin[n])^2))),n]//Timing

V13.3

Sum[(1/((n^3)((Sin[n])^2))),{n,∞}]//Timing

SumConvergence[(1/((n^3)((Sin[n])^2))),n]//Timing

Regularlized sums

Sum[(1/((n^3)((Sin[n])^2))),{n,∞},Regularization->"Abel"]//Timing

Sum[(1/((n^3)((Sin[n])^2))),{n,∞},Regularization->"Borel"]//Timing

Sum[(1/((n^3)((Sin[n])^2))),{n,∞},Regularization->"Cesaro"]//Timing

Sum[(1/((n^3)((Sin[n])^2))),{n,∞},Regularization->"Dirichlet"]//Timing

Table

N[Table[Csc[n],{n,100}],10]

A002485

Convergents[Pi,10]

Numerator[%]

A046947

HighWaterMarks[Abs[Table[Csc[n],{n,10^6}]]]//Timing

N[%[[2,1]]]

Convergents[c_,n_]:=Table[FromContinuedFraction[ContinuedFraction[c,k]],{k,n}]

Convergents[Pi,10]

Numerator[%]

Graphics

data=FoldList[Plus,0,N[Table[(1/((n^3)((Sin[n])^2))),{n,10^4}],100]];

g1=ListPlot[data,Joined->True,PlotStyle->Red,PlotRange->All,AxesLabel->{n,(Subscript[S, 1])[n]},LabelStyle->{Black,FontFamily->"Times",14}]

g2=ListPlot[data,Joined->True,PlotStyle->Red,AxesLabel->{n,(Subscript[S, 1])[n]},LabelStyle->{Black,FontFamily->"Times",14},PlotRange->{30.28,30.32}]

bug(430973): GraphicsRow of ListPlot's cuts of left side of axes tick labels

GraphicsRow[{g1,g2},ImageSize->700]