Menu Close

n-N-suppose-u-n-5sin-1-n-2-1-5-cos-n-n-Prove-that-lim-n-u-n-0-




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
$$\forall{n}\in\mathbb{N}^{\ast} ,\:\mathrm{suppose}\:{u}_{{n}} =\left(\mathrm{5sin}\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{5}}\mathrm{cos}\:{n}\right)^{{n}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{n}\rightarrow+\infty} {\mathrm{lim}}{u}_{{n}} =\mathrm{0} \\ $$
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
$${since}\:{sin}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)\rightarrow\mathrm{0}\Rightarrow\exists{N}\:{such}\:\forall{n}\geqslant{N} \\ $$$$\mathrm{0}\leqslant{sin}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)<\frac{\mathrm{1}}{\mathrm{10}} \\ $$$$\forall{x}\in\mathbb{R}\:\:\:\:−\mathrm{1}\leqslant{cos}\left({x}\right)\leqslant\mathrm{1} \\ $$$$\Rightarrow\forall{n}\geqslant{N} \\ $$$$\mathrm{0}−\frac{\mathrm{1}}{\mathrm{5}}\leqslant\mathrm{5}{sin}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)+\frac{{cos}\left({n}\right)}{\mathrm{5}}\leqslant\frac{\mathrm{5}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{5}}=\frac{\mathrm{7}}{\mathrm{10}} \\ $$$$\forall{n}\geqslant{N} \\ $$$$\:\:\:\:\:−\frac{\mathrm{1}}{\mathrm{5}}\leqslant\mathrm{5}{sin}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)+\frac{{cos}\left({n}\right)}{\mathrm{5}}\leqslant\frac{\mathrm{7}}{\mathrm{10}} \\ $$$$\Rightarrow−\left(\frac{\mathrm{1}}{\mathrm{5}}\right)^{{n}} \leqslant\left(\mathrm{5}{sin}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)+\frac{{cos}\left({n}\right)}{\mathrm{5}}\right)^{{n}} \leqslant\left(\frac{\mathrm{7}}{\mathrm{10}}\right)^{{n}} \\ $$$${since}\:−\left(\frac{\mathrm{1}}{\mathrm{5}}\right)^{{n}} \rightarrow\mathrm{0},\left(\frac{\mathrm{7}}{\mathrm{10}}\right)^{{n}} \rightarrow\mathrm{0} \\ $$$${we}\:{get}\:{our}\:{Ansewr} \\ $$$$ \\ $$$$ \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *