\(\alpha_{n}=2\,\alpha_{n-1}+2^{-2(n-1)}\)
Posted: Wed Oct 18, 2017 11:23 am
For the sequence $\left({\alpha_{n}}\right)_{n\in\mathbb{N}\cup\{0\}}$ of real numbers defined recursively as
\[\alpha_{n}=2\,\alpha_{n-1}+2^{-2(n-1)}\,,\; n\in\mathbb{N}\,,\quad \alpha_0=1\,:\]
\[\alpha_{n}=2\,\alpha_{n-1}+2^{-2(n-1)}\,,\; n\in\mathbb{N}\,,\quad \alpha_0=1\,:\]
- Find the general form of \(\alpha_{n}\).
- Find the values of real number $\beta$ for which the \(\displaystyle\mathop{\lim}\limits_{n\rightarrow{+\infty}}\frac{2^{\alpha_{n}}}{2^{{\beta}^{n}}}\) is a real number.