Question Number 127455 by snipers237 last updated on 29/Dec/20
![Prove that for all n≥1 1)There exist a_n ∈]0,1[ such as sin((1/n))=(1/n)−(1/(6n^3 ))cos((1/n)a_n ) 2) Prove that lim_(n→∞) a_n = (1/(10))](Q127455.png)
$${Prove}\:{that}\:{for}\:{all}\:{n}\geqslant\mathrm{1}\: \\ $$$$\left.\mathrm{1}\left.\right){There}\:{exist}\:\:{a}_{{n}} \in\right]\mathrm{0},\mathrm{1}\left[\:{such}\:{as}\:\:\right. \\ $$$${sin}\left(\frac{\mathrm{1}}{{n}}\right)=\frac{\mathrm{1}}{{n}}−\frac{\mathrm{1}}{\mathrm{6}{n}^{\mathrm{3}} }{cos}\left(\frac{\mathrm{1}}{{n}}{a}_{{n}} \right) \\ $$$$\left.\mathrm{2}\right)\:{Prove}\:{that}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{10}}\: \\ $$