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Question Number 65778 by mathmax by abdo last updated on 03/Aug/19
find lim_(n→+∞) e^(−n^2 ) (n+1)^(n!)
findlimn→+∞e−n2(n+1)n!
Commented by mathmax by abdo last updated on 08/Aug/19
let A_n =e^(−n^2 ) (n+1)^(n!) ⇒A_n =e^(−n^2 ) e^(n!ln(n+1)) = e^(−n^2 +n(n−1)(n−2)!ln(n+1)) =e^(n^2 (−1 +(1−(1/n))(n−2)!ln(n+1))) its clear that lim_(n→+∞) −1+(1−(1/n))(n−2)!ln(n+1) =+∞ ⇒ lim_(n→+∞) A_n =+∞
letAn=e−n2(n+1)n!⇒An=e−n2en!ln(n+1)=e−n2+n(n−1)(n−2)!ln(n+1)=en2(−1+(1−1n)(n−2)!ln(n+1))itsclearthatlimn→+∞−1+(1−1n)(n−2)!ln(n+1)=+∞⇒limn→+∞An=+∞

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