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 Post subject: $\alpha_{n}=2\,\alpha_{n-1}+2^{-2(n-1)}$Posted: Wed Oct 18, 2017 11:23 am
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Joined: Mon Nov 09, 2015 1:36 am
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Location: Ioannina, Greece
For the sequence $\left({\alpha_{n}}\right)_{n\in\mathbb{N}\cup\{0\}}$ of real numbers defined recursively as
$\alpha_{n}=2\,\alpha_{n-1}+2^{-2(n-1)}\,,\; n\in\mathbb{N}\,,\quad \alpha_0=1\,:$
1. Find the general form of $\alpha_{n}$.
2. 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.

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Grigorios Kostakos

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