■ある無限級数(その25)
1/(1−z)=1+z+z^2+・・・
この導関数と積分は
1/(1−z)^2=1+2z+3z^2+・・・
ln(1/(1−z))=z+z^2/2+z^3/3+・・・
(その17)を補足しておきたい.
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z/(1−z)=z+z^2+z^3+・・・
z^n/(1−z^n)=z^n+z^2n+z^3n+・・・
Σz^n/(1−z^n)=Σ(z^n+z^2n+z^3n+・・・)
=Σkz^k(1−z^k+1)(1−z^k+2)・・・
=Σ(1+z^n)/(1−z^n)・z^(n^2)
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