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Question Number 173192 by mathlove last updated on 08/Jul/22

$$\underset{n\to \mathrm{\infty }}{\lim }{n}^2\sqrt{(1-cos\frac{1}{n})\sqrt{(1-cos\frac{1}{n})\sqrt{(1-cos\frac{1}{n})......\mathrm{\infty }}}}=?$$

Answered by Frix last updated on 08/Jul/22

$$u=\sqrt{t\sqrt{t\sqrt{t\sqrt{...}}}}\iff u=\sqrt{tu}\iff u=t$$$$$$$$\underset{n\to +\mathrm{\infty }}{\lim }\left[n^2(1-\cos \frac{1}{n})\right]=$$$$=\underset{k\to 0^{+}}{\lim }\left[\frac{1-\cos k}{k^2}\right]=$$$$=\underset{k\to 0^{+}}{\lim }\left[\frac{\sin k}{2k}\right]=$$$$=\underset{k\to 0^{+}}{\lim }\left[\frac{\cos k}{2}\right]=$$$$=\frac{1}{2}$$

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