Question Number 104174 by Dwaipayan Shikari last updated on 19/Jul/20

Π_(n=1) ^∞ ((n/(n+1)))^2

Answered by OlafThorendsen last updated on 19/Jul/20

S_n  = Π_(k=1) ^n ((k/(k+1)))^2   S_n  = ((1/2))^2 ((2/3))^2 ((3/4))^2 ...((n/(n+1)))^2   S_n  = (1/((n+1)^2 ))  lim_(n→∞) S_n  = 0

Answered by mathmax by abdo last updated on 19/Jul/20

let S_n =Π_(k=1) ^n  (k^2 /((k+1)^2 )) ⇒ln(S_n ) =Σ_(k=1) ^n ln((k^2 /((k+1)^2 ))) =2Σ_(k=1) ^n { ln(k)−ln(k+1)}  =2 {ln1−ln(2)+ln(2)−ln(3)+...+ln(n)−ln(n+1)}  =−2ln(n+1)→−∞ ⇒lim_(n→+∞) ln(S_n ) =0 ⇒lim_(n→+∞) S_n =0