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 Post subject: Limit of a sequencePosted: Wed Oct 25, 2017 8:02 pm
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Joined: Sat Nov 07, 2015 6:12 pm
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Location: Larisa
Define the sequence $\{k_n\}_{n \in \mathbb{N}}$ recursively as follows

$$k_0 = \frac{1}{\sqrt{2}} \quad , \quad k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}{1+{\sqrt {1-k_{n}^{2}}}}}$$

Evaluate the limit

$$\ell = \lim_{n \rightarrow + \infty} \left(\frac{4}{k_{n+1}}\right)^{2^{-n}}$$

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