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Question Number 116166 by Ar Brandon last updated on 01/Oct/20

∀n∈N^∗ , suppose u_n =(5sin(1/n^2 )+(1/5)cos n)^n Prove that lim_(n→+∞) u_n =0

∀n∈N∗,supposeun=(5sin1n2+15cosn)nProvethatlimn→+∞un=0

Answered by mindispower last updated on 01/Oct/20

since sin((1/n^2 ))→0⇒∃N such ∀n≥N 0≤sin((1/n^2 ))<(1/(10)) ∀x∈R −1≤cos(x)≤1 ⇒∀n≥N 0−(1/5)≤5sin((1/n^2 ))+((cos(n))/5)≤(5/(10))+(1/5)=(7/(10)) ∀n≥N −(1/5)≤5sin((1/n^2 ))+((cos(n))/5)≤(7/(10)) ⇒−((1/5))^n ≤(5sin((1/n^2 ))+((cos(n))/5))^n ≤((7/(10)))^n since −((1/5))^n →0,((7/(10)))^n →0 we get our Ansewr

sincesin(1n2)→0⇒∃Nsuch∀n⩾N0⩽sin(1n2)<110∀x∈R−1⩽cos(x)⩽1⇒∀n⩾N0−15⩽5sin(1n2)+cos(n)5⩽510+15=710∀n⩾N−15⩽5sin(1n2)+cos(n)5⩽710⇒−(15)n⩽(5sin(1n2)+cos(n)5)n⩽(710)nsince−(15)n→0,(710)n→0wegetourAnsewr

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