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Question Number 30755 by abdo imad last updated on 25/Feb/18

find lim_(n→∞) ^n (√((1+(1/n))(1+(2/n))...(1+(n/n)) ))

findlimn→∞n(1+1n)(1+2n)...(1+nn)

Commented by abdo imad last updated on 28/Feb/18

let put A_n =^n (√((1+(1/n))(1+(2/n))....(1+(n/n)))) we have ln(A_n )= (1/n) ln(Π_(k=1) ^n (1+(k/n)))=(1/n) Σ_(k=1) ^n ln(1+(k/n))so ln(A_n ) is a Rieman sum and lim_(n→∞) ln(A_n ) = ∫_0 ^1 ln(1+x)dx= ∫_1 ^2 ln(t)dt=[tlnt −t]_1 ^2 =2ln2−2 +1=2ln2 −1 ⇒lim_(n→∞) A_n = e^(−1) e^(ln(4)) =(4/e) .

letputAn=n(1+1n)(1+2n)....(1+nn)wehaveln(An)=1nln(∏k=1n(1+kn))=1n∑k=1nln(1+kn)soln(An)isaRiemansumandlimn→∞ln(An)=∫01ln(1+x)dx=∫12ln(t)dt=[tlnt−t]12=2ln2−2+1=2ln2−1⇒limn→∞An=e−1eln(4)=4e.

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