■ある無限級数(その38)

  Hx=Σ{1/n-1/(n+x)}

 x=p/qのとき,

Hx=q/p-π/2・cotpπ/q-log2q+2Σcos2pkπ/q・logsinkπ/q  (0<k<q/2)

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  H1/2=2-2log2

  H1/3=3-π/2√3-3/2・log3

  H2/3=3/2+π/2√3-3/2・log3

  H1/4=4-π/2-3log2

  H3/4=4/3+π/2-3log2

  H1/5=5-π/2・φ{(2+φ)/5}^1/2-1/2・(3-φ)log5-(φ-1/2)log(2+φ)

  H2/5=5/2-π/2・1/φ(2+φ)^1/2-1/2・(2+φ)log5+(φ-1/2)log(2+φ)

  H3/5=5/3+π/2・1/φ(2+φ)^1/2-1/2・(2+φ)log5+(φ-1/2)log(2+φ)

  H4/5=5/4+π/2・φ{(2+φ)/5}^1/2-1/2・(3-φ)log5-(φ-1/2)log(2+φ)

  H1/6=6-π√3/2-2log2-3/2・log3

  H5/6=6/5+π√3/2-2log2-3/2・log3

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