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Monday, December 5, 2016

Pseudo sum

Let $\alpha, \beta $ be positive irrational numbers such that $\displaystyle \frac{1}{\alpha} + \frac{1}{\beta}=1$. Evaluate the (pseudo) sum:


$$\sum_{n=1}^{\infty} \left(\frac{1}{\lfloor n\alpha\rfloor^2}+\frac{1}{\lfloor n\beta\rfloor^2}\right)$$

Solution

There is this theorem called Beatty's theorem that our solution is based on. In brief, it states that for positive irrational numbers $\alpha, \beta$ with $\displaystyle \frac{1}{\alpha}+\frac{1}{\beta}=1$ the sequences $[\lfloor \alpha\rfloor, \lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor, \dotsc$ and $\lfloor \beta\rfloor, \lfloor 2\beta\rfloor, \lfloor 3\beta\rfloor, \dotsc$ are complementary. (i.e. disjoint and their union is $\mathbb{N}$). Thus our sum is nothing else than $\zeta(2)=\frac{\pi^2}{6}$. This justifies the title "Pseudo".


 

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