prove-that-lim-x-n-2-1-cos-1-n-1-cos-1-n-1-cos-1-n-1-2-

Question Number 82160 by M±th+et£s last updated on 18/Feb/20

p r o v e t h a t

\underset{x \to \infty}{l i m} n^{2} \sqrt{(1 - c o s (\frac{1}{n}) \sqrt{(1 - c o s \frac{1}{n}) \sqrt{(1 - c o s \frac{1}{n}) \dots}}} = \frac{1}{2}

Answered by MJS last updated on 18/Feb/20

w = \sqrt{(1 - \cos \frac{1}{n}) \sqrt{(1 - \cos \frac{1}{n}) \sqrt{\dots}}}

w^{2} = (1 - \cos \frac{1}{n}) w \Rightarrow w = 0 \lor w = 1 - \cos \frac{1}{n}

\dots = \lim_{n \to \infty} n^{2} (1 - \cos \frac{1}{n})

let n = \frac{1}{k}

\lim_{k \to 0} \frac{1 - \cos k}{k^{2}} = \lim_{k \to 0} \frac{\frac{d^{2}}{{d k}^{2}} [1 - \cos k]}{\frac{d^{2}}{{d k}^{2}} [k^{2}]} =

= \lim_{k \to 0} \frac{\cos k}{2} = \frac{1}{2}

Answered by mr W last updated on 18/Feb/20

1 - \cos \frac{1}{n} = 2 \sin^{2} \frac{1}{2 n}

\underset{n \to \infty}{l i m} n^{2} \sqrt{(1 - c o s (\frac{1}{n}) \sqrt{(1 - c o s \frac{1}{n}) \sqrt{(1 - c o s \frac{1}{n}) \dots}}}

= \underset{n \to \infty}{l i m} n^{2} {(2 \sin^{2} \frac{1}{2 n})}^{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots}

= \underset{n \to \infty}{l i m} n^{2} {(2 \sin^{2} \frac{1}{2 n})}^{\frac{1}{2} \times \frac{1}{1 - \frac{1}{2}}}

= \underset{n \to \infty}{l i m} n^{2} (2 \sin^{2} \frac{1}{2 n})

= \underset{n \to \infty}{l i m} \frac{1}{2} {(\frac{\sin \frac{1}{2 n}}{\frac{1}{2 n}})}^{2}

= \frac{1}{2} \times 1 = \frac{1}{2}

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