# Series with inverse Fibonacci products

Thanks for stopping by to look at my question.

I'm trying to show that $R$ is irrational in $$R:= \dfrac{1}{F_1}+\dfrac{1}{F_2\cdot F_1}+\dfrac{1}{F_3\cdot F_2 \cdot F_1}+\dfrac{1}{F_4\cdot F_3\cdot F_2 \cdot F_1}+\cdots$$ where $F_n$ are the Fibonacci numbers and $R<3$ and $F_{n+k}>F_k$ for $n>0$.

How should I approach this? Where is this series from?

• My first thought is that we might be able to rewrite this as a product, and then show that any rational denominator would be arbitrarily large. As to where the series is from, you should perhaps share where you found it or if you thought of it yourself. – hardmath Nov 15 '16 at 17:44
• Thanks, @hardmath. No, I didn't think of it, it's listed as a bonus question on a homework assignment. – jbrow35 Nov 15 '16 at 17:46