■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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