Limit of a sequence
Posted: Wed Oct 25, 2017 8:02 pm
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
$$\ell = \lim_{n \rightarrow + \infty} \left(\frac{4}{k_{n+1}}\right)^{2^{-n}}$$
$$k_0 = \frac{1}{\sqrt{2}} \quad , \quad k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}{1+{\sqrt {1-k_{n}^{2}}}}}$$
Evaluate the limit
$$\ell = \lim_{n \rightarrow + \infty} \left(\frac{4}{k_{n+1}}\right)^{2^{-n}}$$