BALKAN-SL 2010 A2

Let the sequence $(a_n)_{n \in \mathbb{N}}$ , where $\mathbb{N}$ denote the set of natural numbers, is given with $a_1=2$ and $a_{n+1}$ $=$ $a_n^2$ $-$ $a_n+1$ . Find the minimum real number $L$ , such that for every $k$ $\in$ $\mathbb{N}$ \begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*}