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Question Number 41705 by abdo.msup.com last updated on 11/Aug/18

find nsture of the serie Σ u_n   with u_n =^n (√(n/(n+1)))   −1

$${find}\:{nsture}\:{of}\:{the}\:{serie}\:\Sigma\:{u}_{{n}} \\ $$$${with}\:{u}_{{n}} =^{{n}} \sqrt{\frac{{n}}{{n}+\mathrm{1}}}\:\:\:−\mathrm{1} \\ $$

Commented by maxmathsup by imad last updated on 12/Aug/18

we have  u_n = (1−(1/(n+1)))^(1/n)  −1  = e^((1/n)ln(1−(1/(n+1))))  −1  but we have  ln^′ (1−u) =((−1)/(1−u)) =−Σ_(n=0) ^∞  u^n  ⇒ln(1−u) =−Σ_(n=0) ^∞   (u^(n+1) /(n+1))  =− Σ_(n=1) ^∞  (u^n /n)  ⇒ln(1−u) =−u −(u^2 /2) +o(u^3 ) ⇒  ln(1−(1/(n+1))) =−(1/(n+1)) −(1/(2(n+1)^2 )) +o((1/((n+1)^3 ))) ⇒  (1/n)ln(1−(1/(n+1))) =−(1/(n(n+1))) −(1/(2n(n+1)^2 )) +o((1/n^2 )) ⇒  u_n ∼ e^(−(1/(n(n+1))))  −1 ∼ 1−(1/(n(n+1))) −1 ⇒u_n ∼ −(1/(n(n+1))) but the serie  Σ  (1/(n(n+1))) converges ⇒ Σ u_n  fonverges.

$${we}\:{have}\:\:{u}_{{n}} =\:\left(\mathrm{1}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\right)^{\frac{\mathrm{1}}{{n}}} \:−\mathrm{1}\:\:=\:{e}^{\frac{\mathrm{1}}{{n}}{ln}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\right)} \:−\mathrm{1}\:\:{but}\:{we}\:{have} \\ $$$${ln}^{'} \left(\mathrm{1}−{u}\right)\:=\frac{−\mathrm{1}}{\mathrm{1}−{u}}\:=−\sum_{{n}=\mathrm{0}} ^{\infty} \:{u}^{{n}} \:\Rightarrow{ln}\left(\mathrm{1}−{u}\right)\:=−\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{u}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}} \\ $$$$=−\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{u}^{{n}} }{{n}}\:\:\Rightarrow{ln}\left(\mathrm{1}−{u}\right)\:=−{u}\:−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:+{o}\left({u}^{\mathrm{3}} \right)\:\Rightarrow \\ $$$${ln}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\right)\:=−\frac{\mathrm{1}}{{n}+\mathrm{1}}\:−\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\:+{o}\left(\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} }\right)\:\Rightarrow \\ $$$$\frac{\mathrm{1}}{{n}}{ln}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\right)\:=−\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}\:−\frac{\mathrm{1}}{\mathrm{2}{n}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\:+{o}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)\:\Rightarrow \\ $$$${u}_{{n}} \sim\:{e}^{−\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}} \:−\mathrm{1}\:\sim\:\mathrm{1}−\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}\:−\mathrm{1}\:\Rightarrow{u}_{{n}} \sim\:−\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}\:{but}\:{the}\:{serie} \\ $$$$\Sigma\:\:\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}\:{converges}\:\Rightarrow\:\Sigma\:{u}_{{n}} \:{fonverges}. \\ $$

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