This short article is inspired by Lesko’s[1] article on CMJ.
ln(3)1+-++-+...
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ln(4)1++-+++-+...
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ln(5)1+++-+++-+...
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...
The key to generate the series is to use a u-substitution for where . Thus the integration from to is mapped to to .
du
u
u1+x++...+
2
x
n-1
x
x0
x1
u1
un
u[n_]:=Total[x^Range[0,n-1]]
For instance :
n5
D[u[5],x]
u[5]
Out[]=
1+2x+3+4
2
x
3
x
1+x+++
2
x
3
x
4
x
Integrate the expression to show it is what we need
Integrate[%,{x,0,1}]
Out[]=
Log[5]
Use Apply[List,...] to convert the rational quotient function to Taylor series and extract each term from the series:
List@@Series,{x,0,20}//Normal
D[u[5],x]
u[5]
Out[]=
{1,x,,,-4,,,,,-4,,,,,-4,,,,,-4,}
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Item-wise integration generates the reduced fraction we need in the form for at the beginning:
ln5
(Integrate[#,{x,0,1}]&/@%)
Out[]=
1,,,,-,,,,,-,,,,,-,,,,,-,
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ComplexPlot,{x,1.4},ColorFunction"ShiftedCyclicLogAbs",ImagePadding{{20,3},{15,3}}
D[u[5],x]
u[5]
Out[]=
Create a gallery for several ’s:
n
ImageAssemble@PartitionTableComplexPlot,{x,1.4},ColorFunction"ShiftedCyclicLogAbs",Frame->False,FrameTicksNone,{i,2,10},3,3
D[u[i],x]
u[i]
Out[]=
Reference
Reference
[1] Lesko, James. “A Series for Ln k.” The College Mathematics Journal 32, no. 2 (2001): 119–22. https://doi.org/10.2307/2687117.