Favard constant

Remarks

The series converges for every $r \ge 0$: for $r = 0$ it is an alternating series, and for $r \ge 1$ it converges absolutely.

Example: $K_0 = \frac{4}{\pi} \left(1 - \frac{1}{3} + \frac{1}{5} - \cdots\right) = \frac{4}{\pi} \cdot \frac{\pi}{4} = 1$. Likewise $K_1 = \frac{4}{\pi} \sum_{k \ge 0} \frac{1}{(2k+1)^2} = \frac{4}{\pi} \cdot \frac{\pi^2}{8} = \frac{\pi}{2}$. Further values are $K_2 = \pi^2/8$ and $K_3 = \pi^3/24$.

Non-example: $\frac{4}{\pi} \sum_{k \ge 0} \frac{1}{2k + 1}$ is not a Favard constant; it lacks the signs of the case $r = 0$ and the series diverges.

The even-indexed constants increase and the odd-indexed constants decrease, and both tend to $4/\pi$. So $1 = K_0 < K_2 < K_4 < \cdots < \frac{4}{\pi} < \cdots < K_3 < K_1 = \frac{\pi}{2}$.