Convergent Series

Author

Eric W. Weisstein
November 23, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Series

(∑_(n=2)^∞)(1/(n log(n)))=∞

V6.0

Integrate[(1/(n Log[n])),{n,2,∞}]

Limit[(1/(n Log[n])),n->∞]

Sum[(1/(n Log[n])),{n,2,∞}]

NSum[(1/(n Log[n])),{n,2,∞}]

NSum[(1/(n Log[n])),{n,2,∞},Method->Fit,PrecisionGoal->10]

NSum[(1/(n Log[n])),{n,2,∞},Method->NIntegrate,PrecisionGoal->10]

V7

Sum[(1/(n Log[n])),{n,2,∞}]

SumConvergence[(1/(n Log[n])),n]

(∑_(n=2)^∞)(1/(n (log^2)(n)))=2....

Integral Test

Limit[(1/(n ((Log[n])^2))),n->∞]

Integrate[(1/(n ((Log[n])^2))),{n,2,∞}]

Plots

ListPlot[FoldList[Plus,0,Table[((N[1,100])/(n ((Log[n])^2))),{n,2,1000}]],Joined->True,PlotStyle->Red]

Symbolic

Sum[(1/(n ((Log[n])^2))),{n,2,∞}]

NSum

NSum[(1/(n ((Log[n])^2))),{n,2,∞},WorkingPrecision->300,PrecisionGoal->15,Method->Fit,NSumTerms->1000,NSumExtraTerms->1000]

NSum[(1/(n ((Log[n])^2))),{n,2,∞},WorkingPrecision->30,PrecisionGoal->15,Method->Integrate]//Timing

Table Sums

Total[Table[((1`20)/(n ((Log[n])^2))),{n,2,10^5}]]//Timing

Total[Table[((1`20)/(n ((Log[n])^2))),{n,2,10^6}]]//Timing

Total[Table[((1`20)/(n ((Log[n])^2))),{n,2,10^7}]]//Timing

Total[Table[((1`20)/(n ((Log[n])^2))),{n,2,10^8}]]//Timing

V7

Sum[(1/(n ((Log[n])^2))),{n,2,∞}]

SumConvergence[(1/(n ((Log[n])^2))),n]

(∑_(k=3)^∞)(1/(k log(k) (log^2)(log(k))))==38.43...

Limit Test

Limit[(1/(k Log[k]((Log[Log[k]])^2))),k->∞]

Integrate[(1/(k Log[k]((Log[Log[k]])^2))),{k,2,∞}]

Integrate[(1/(k Log[k]((Log[Log[k]])^2))),{k,3,∞}]

Plots

Plot[(1/(k Log[k]((Log[Log[k]])^2))),{k,2,3},PlotStyle->Red]

Plot[(1/(k Log[k]((Log[Log[k]])^2))),{k,3,100},PlotStyle->Red]

Symbolic

Sum[(1/(k Log[k]((Log[Log[k]])^2))),{k,3,n}]

Sum[(1/(k Log[k]((Log[Log[k]])^2))),{k,3,∞}]

NSum

NSum[(1/(k Log[k]((Log[Log[k]])^2))),{k,3,∞},VerifyConvergence->True]

NSum[(1/(k Log[k]((Log[Log[k]])^2))),{k,3,∞},WorkingPrecision->200,PrecisionGoal->20,Method->Fit,NSumTerms->1000,NSumExtraTerms->2000]//Timing

NSum[(1/(k Log[k]((Log[Log[k]])^2))),{k,3,∞},WorkingPrecision->200,PrecisionGoal->10,Method->Integrate]//Timing

Table Sums

Sum[((1`20)/(k Log[k]((Log[Log[k]])^2))),{k,3,10^4}]//Timing

Total[Table[((1`20)/(k Log[k]((Log[Log[k]])^2))),{k,3,10^4}]]//Timing

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

Sum[((1`20)/(k Log[k]((Log[Log[k]])^2))),{k,3,10^5}]//Timing

Sum[((1`20)/(k Log[k]((Log[Log[k]])^2))),{k,3,10^6}]//Timing

Total[Table[((1`20)/(k Log[k]((Log[Log[k]])^2))),{k,3,10^7}]]//Timing

Total[Table[((1`20)/(k Log[k]((Log[Log[k]])^2))),{k,3,10^8}]]//Timing

V7

Sum[(1/(k Log[k]((Log[Log[k]])^2))),{k,3,∞}]

SumConvergence[(1/(k Log[k]((Log[Log[k]])^2))),k]

(∑_(k=3)^∞)(1/(k log(k) log(log(k))))==∞

Limit[(1/(k Log[k]Log[Log[k]])),k->∞]

Integrate[(1/(k Log[k]Log[Log[k]])),{k,3,∞}]

Sum[(1/(k Log[k]Log[Log[k]])),{k,3,∞}]

Sum[(1/k),{k,∞}]

V7

Sum[(1/(k Log[k]Log[Log[k]])),{k,3,∞}]

SumConvergence[(1/(k Log[k]Log[Log[k]])),k]