I'm looking for a proof for $\frac 14=\sum_{k=2}^{\infty}\frac{\left(\frac{p_k-1}{2}\right)}{P(p_k)}$

Question

I'm interested in Primorial numeral system (primoradic, see stub OEIS). In that system, you can define "primorial fractions" as fractions which can be writen as $$ \frac{{a_1 }}{2} + \frac{{a_2 }}{6} + \frac{{a_3 }}{{30}} + \cdots + \frac{{a_n }}{{P(p_n )}} $$ with $a_1$ in $\left\{ {0,1} \right\}$, $a_2$ in $\left\{ {0,1,2} \right\}$, $a_3$ in $\left\{ {0,1,2,3,4} \right\}$, $\ldots$ , $a_n$ in $\left\{ {0,1,2,\ldots,p_n -1} \right\}$. For example, $\frac{7}{10}=\frac{21}{30}=\frac{15+5+1}{30}=\frac{15}{30}+\frac{5}{30}+\frac{1}{30}=\frac{1}{2}+\frac{1}{6}+\frac{1}{30}$ is a primorial fraction. The presentation of the formula, which may surprise, is very important for me.It is interesting to approach other fractions or e=2.71828... for example with primorial fractions. It is not a simple question of number(here the number 0.25), nor a question of system of the world BUT it is a question of SYSTEM. As far as I can judge, $\frac{1}{4}$=$\frac{1}{2}$x$\frac{1}{2}$ is not a primorial fraction even if $\frac{1}{2}$ IS a primorial fraction. If you draw ("si vous faites un dessin, un schéma"), the formula becomes clear. But I had to prove the development in series given. For $k=2$ to $9$, you obtain $0.249999997758\ldots$. Sorry for my very bad English (French). Cordialement, Stéphane Jaouen P.S. : I propose an extension for the notations adopted in stub OEIS concerning Primoradic number system : for example, 2+$\frac{1}{2}$+$\frac{1}{6}$+$\frac{1}{30}$=2.7=[2,1:1:1]. So that [2,1:1:1]<e. I don't know if it has already be done elsewhere.

Answer 1

Notice that

$$ \frac{p_k}{P(p_k)}= \frac{1}{P(p_{k-1})}.$$ Therefor, we can telescope this series to find that

$$\sum_{k=2}^n \frac{\frac{p_k-1}{2}}{P(p_k)}=\frac{1}{2} \sum_{k=2}^n \big{(}\frac{1}{P(p_{k-1})}-\frac{1}{P(p_k)}\big{)}=1/4-\frac{1}{2P(p_n)} \to 1/4 .$$

as $n \to \infty$.