Floor and Ceiling Series (I)

Question

Prove or disprove:

1) $$\sum_{n=2}^{\infty}{\frac{1}{\lfloor n^2/2 \rfloor}} = \frac{1+\zeta(2)}{2}$$

2) $$\sum_{n=1}^{\infty}{\frac{1}{\lceil\ n^2/2 \rceil}} = \frac{\zeta(2)}{2} + \frac{\pi}{2} \tanh(\frac{\pi}{2})$$

Answer 1

Just split the series into even and odd terms. The first one:

$$\sum_{n=2}^\infty \frac{1}{\lfloor n^2/2 \rfloor}=\sum_{n=1}^\infty \frac{1}{2n^2}+\sum_{n=1}^\infty \frac{1}{2n^2+2n}$$ The second: $$\sum_{n=2}^\infty \frac{1}{\lceil n^2/2 \rceil}=\sum_{n=1}^\infty \frac{1}{2n^2}+\sum_{n=1}^\infty \frac{1}{2n^2+2n+1}$$

The first term is simply $\zeta(2)/2$. The second term in the first expression is a simple telescoping sum. The second term of second expression is a bit trickier. I'm trying partial fractions, I'll post when I find an elegant solution.