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Question Number 35057 by math khazana by abdo last updated on 14/May/18

let u_n =((n/(n+1)))^(1/n) −1 find the nature of Σ_(n≥0) u_n

letun=(nn+1)1n−1findthenatureof∑n⩾0un

Commented by prof Abdo imad last updated on 16/May/18

we have u_n = (1−(1/(n+1)))^n −1 =e^(nln(1−(1/(n+1)))) −1 but 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) =−(u +(u^2 /2) +.....)⇒ ln(1−(1/(n+1))) =−(1/(n+1)) −(1/(2(n+1)^2 )) +o((1/((n+1)^3 )))⇒ nln(1−(1/(n+1))) =((−n)/(n+1)) −(n/((n+1)^2 )) +o( (1/n^2 ))→−1(n→+∞) ⇒lim_(n→+∞) u_n =(1/e) ≠0 so Σ u_n diverges .

wehaveun=(1−1n+1)n−1=enln(1−1n+1)−1butln′(1−u)=−11−u=−∑n=0∞un⇒ln(1−u)=−∑n=0∞un+1n+1=−∑n=1∞unn=−(u+u22+.....)⇒ln(1−1n+1)=−1n+1−12(n+1)2+o(1(n+1)3)⇒nln(1−1n+1)=−nn+1−n(n+1)2+o(1n2)→−1(n→+∞)⇒limn→+∞un=1e≠0soΣundiverges.

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