I discovered an interesting series that seems to yield rational numbers related to π:
$$ \sum_{k=0}^∞\frac{m}{\Pi_{i∈I}(4k+i)}=\left|\frac{p}{q}-π\right| $$
I don't know if it's a coincidence or if the series can approximate any rational number, I found the following decimal approximation:
$$ \begin{aligned} π-3&=\sum_{k=0}^∞\frac{24}{(4k+2)(4k+3)(4k+5)(4k+6)}>0\\ 4-π&=\sum_{k=0}^∞\frac{5040}{(4k+1)(4k+2)(4k+3)(4k+9)(4k+10)(4k+11)}>0\\ π-\frac{31}{10}&=\quad?\\ \frac{32}{10}-π&=\sum_{k=0}^∞\frac{48}{(4k+3)(4k+5)(4k+7)(4k+9)}>0\\ π-\frac{314}{100}&=\sum_{k=0}^∞\frac{9072}{(4k+2)(4k+5)(4k+6)(4k+7)(4k+9)(4k+10)(4k+11)(4k+14)}>0\\ \frac{315}{100}-π&=\sum_{k=0}^∞\frac{5040}{(4k+2)(4k+3)(4k+4)(4k+5)(4k+7)(4k+\phantom{0}8)(4k+\phantom{0}9)(4k+10)}>0\\ \end{aligned} $$
Can we find the constant $m$ and the set of coefficients $I$ such that the series value is $π-\frac{31}{10}$
Update 1
I expanded my search and found the following linear series:
$$ \pi-\frac{31}{10}=\frac{16}{5}\sum_{k=0}^{\infty}\frac{2-13k}{(4k+1)(4k+3)(4k+5)(4k+7)} $$
For constant series, all combinations with coefficients less than 20 fail to give $\pi-\frac{31}{10}$.