Question 37
Let $\left(F_{n}\right)_{n \in \mathbb{N}^{*}}$ be the Fibonacci sequence so that: $F_{1}=1$, $F_{2}=1$, $F_{n+1}=F_{n}+F_{n-1}$ for every positive integer $n \geq 2$. Show that $F_{n}\lt 3^{n / 2}$ and calculate the limit
$$
\lim_{n \rightarrow \infty}\left(\frac{F_{1}}{2}+\frac{F_{2}}{2^{2}}+\ldots+\frac{F_{n}}{2^{n}}\right).
$$
$$
\lim_{n \rightarrow \infty}\left(\frac{F_{1}}{2}+\frac{F_{2}}{2^{2}}+\ldots+\frac{F_{n}}{2^{n}}\right).
$$