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lim-n-n-2-1-cos-1-n-1-cos-1-n-1-cos-1-n-




Question Number 173192 by mathlove last updated on 08/Jul/22
lim_(n→∞)  n^2 (√((1−cos(1/n))(√((1−cos(1/n))(√((1−cos(1/n))......∞))))))=?
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{n}^{\mathrm{2}} \sqrt{\left(\mathrm{1}−{cos}\frac{\mathrm{1}}{{n}}\right)\sqrt{\left(\mathrm{1}−{cos}\frac{\mathrm{1}}{{n}}\right)\sqrt{\left(\mathrm{1}−{cos}\frac{\mathrm{1}}{{n}}\right)……\infty}}}=? \\ $$
Answered by Frix last updated on 08/Jul/22
u=(√(t(√(t(√(t(√(...)))))))) ⇔ u=(√(tu)) ⇔ u=t    lim_(n→+∞) [n^2 (1−cos (1/n))] =  =lim_(k→0^+ ) [((1−cos k)/k^2 )] =  =lim_(k→0^+ ) [((sin k)/(2k))] =  =lim_(k→0^+ ) [((cos k)/2)] =  =(1/2)
$${u}=\sqrt{{t}\sqrt{{t}\sqrt{{t}\sqrt{…}}}}\:\Leftrightarrow\:{u}=\sqrt{{tu}}\:\Leftrightarrow\:{u}={t} \\ $$$$ \\ $$$$\underset{{n}\rightarrow+\infty} {\mathrm{lim}}\left[{n}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{cos}\:\frac{\mathrm{1}}{{n}}\right)\right]\:= \\ $$$$=\underset{{k}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[\frac{\mathrm{1}−\mathrm{cos}\:{k}}{{k}^{\mathrm{2}} }\right]\:= \\ $$$$=\underset{{k}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[\frac{\mathrm{sin}\:{k}}{\mathrm{2}{k}}\right]\:= \\ $$$$=\underset{{k}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left[\frac{\mathrm{cos}\:{k}}{\mathrm{2}}\right]\:= \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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