Question Number 148753 by Jonathanwaweh last updated on 30/Jul/21

Answered by mathmax by abdo last updated on 31/Jul/21

A_n =Π_(k=2) ^n e(1−(1/k^2 ))^k^2 ⇒A_n =e^(n−1) Π_(k=2) ^n (1−(1/k^2 ))^k^2 ⇒log(A_n )=n−1+Σ_(k=2) ^n k^2 log(1−(1/k^2 )) log^′ (1−u) =−(1/(1−u))=−(1+u+o(u)) ⇒ log(1−u)=−u−(u^2 /2)+o(u^2 ) ⇒ log(1−(1/k^2 ))=−(1/k^2 )−(1/(2k^4 ))+o((1/k^4 )) ⇒ k^2 log(1−(1/k^2 ))=−1−(1/(2k^2 ))+o((1/k^2 )) ⇒ logA_n ∼n−1+Σ_(k=2) ^n (−1−(1/(2k^2 ))) +o((1/n^2 )) ⇒ logA_n ∼n−1−(n−1)−(1/2)Σ_(k=2) ^n (1/k^2 )+o((1/n^2 )) =−(1/2)(Σ_(k=1) ^n (1/k^2 )−1)=(1/2)−(1/2)ξ_n (2) ⇒ lim_(n→+∞) logA_n =(1/2)−(π^2 /(12)) ⇒ lim_(n→+∞) A_n =e^((1/2)−(π^2 /(12))) =(√e)×e^(−(π^2 /(12)))