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Question Number 82445 by mathmax by abdo last updated on 21/Feb/20
calculate Σ_(n=2) ^∞ ((ξ(n)−1)/n) with ξ(x) =Σ_(n=1) ^∞ (1/n^x ) (x>1)
calculate∑n=2∞ξ(n)−1nwithξ(x)=∑n=1∞1nx(x>1)
Answered by mind is power last updated on 21/Feb/20
=Σ_(n≥2) (1/n).Σ_(m≥2) (1/m^n )=Σ_(m≥2) .Σ_(n≥2) (1/n).((1/m))^n =Σ_(m≥2) .[−ln(1−(1/m))−(1/m)] =Σ_(m≥2) [−ln(m−1)+ln(m)−(1/m)] =lim_(x→∞) .Σ_(m=2) ^x −ln(m−1)+ln(m)−(1/m)] =lim_(x→∞) ln(x)−Σ_(m=2) ^x (1/m) =−lim_(x→∞) {Σ_(m=2) ^x (1/m)−ln(x)} =−lim_(x→∞) {Σ_(m=1) ^x (1/m)−ln(x)−1}=−{γ−1}=1−γ γ Euler mechorini constent
=∑n⩾21n.∑m⩾21mn=∑m⩾2.∑n⩾21n.(1m)n=∑m⩾2.[−ln(1−1m)−1m]=∑m⩾2[−ln(m−1)+ln(m)−1m]=limx→∞.∑xm=2−ln(m−1)+ln(m)−1m]=limx→∞ln(x)−∑xm=21m=−limx→∞{∑xm=21m−ln(x)}=−limx→∞{∑xm=11m−ln(x)−1}=−{γ−1}=1−γγEulermechoriniconstent

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