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> Prove that $$ \sum \limits\_{k=1}^{2n+1} (-1)^{k+1} F\_k \: \text{arccot} F\_k < \frac{\pi+1-\sqrt{5}}{2}$$ holds for all $n \in \mathbb{N}$.
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> (The Fibonacci sequence, defined by the recurrence $F\_1 = F\_2 = 1$ and $\forall n \in \mathbb{N},$ $F\_{n+2} = F\_{n+1} + F\_n$)
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**My work**. I proved that the sequence $a\_n=\sum \limits\_{k=1}^{2n+1} (-1)^{k+1} F\_k \: \text{arccot} F\_k \;$ is increasing. Therefore inequality cannot be proved by induction.