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Question Number 82447 by mathmax by abdo last updated on 21/Feb/20
calculate Σ_(n=2) ^∞ (((−1)^n )/n)ξ(n)
calculate∑n=2∞(−1)nnξ(n)
Answered by mind is power last updated on 21/Feb/20
=Σ_(n≥2) (((−1)^n )/n).Σ_(m≥1) (1/m^n ) =Σ_(m≥1) Σ_(n≥2) (((−1)^n )/(nm^n )) ln(1+x)=Σ_(k≥1) (((−1)^(k−1) x^k )/k) ⇒Σ_(n≥1) (((−1)^n )/n).((1/m))^n =−Σ_(n≥1) (((−1)^(n−1) )/n).((1/m))^n =−ln(1+(1/m)) ⇒Σ_(n≥2) (((−1)^n )/n)((1/m))^n =−ln(1+(1/m))+(1/m) Σ_(m≥1) {−ln(1+(1/m))+(1/m)}=lim_(x→∞) {−ln(1+(1/m))+(1/m)} =lim_(x→∞) Σ_(m=1) ^x {(1/m)}−ln(1+x)=γ
=∑n⩾2(−1)nn.∑m⩾11mn=∑m⩾1∑n⩾2(−1)nnmnln(1+x)=∑k⩾1(−1)k−1xkk⇒∑n⩾1(−1)nn.(1m)n=−∑n⩾1(−1)n−1n.(1m)n=−ln(1+1m)⇒∑n⩾2(−1)nn(1m)n=−ln(1+1m)+1m∑m⩾1{−ln(1+1m)+1m}=limx→∞{−ln(1+1m)+1m}=limx→∞∑xm=1{1m}−ln(1+x)=γ
Commented by mathmax by abdo last updated on 21/Feb/20
thank you sir .
thankyousir.
Commented by mind is power last updated on 21/Feb/20
withe pleasur
withepleasur

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