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$$\sum_{n=1}^\infty \frac{1}{p_n\#} = \frac{1}{2}+\frac{1}{2\times3}+\frac{1}{2\times3\times5}+\dots$$

where $p_n\#$ is the nth Primorial.

Does this sum approaches some known value or constant and do they have a name for it?
I'm also interested in the value for the alternating series which is

$$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{p_n\#} = \frac{1}{2}-\frac{1}{2\times3}+\frac{1}{2\times3\times5}-\dots$$

I have tried finding it in google but nothing seems to pop up. If so I would like to see this calculated to a few decimal places , because I can't find a program to find the an infinite sum base on primorial.

Edit: Is there any literature,papers or study of these 2 series and similar to these series ?

  • 1
    The decimal expansion of the first constant has a record in OEIS: https://oeis.org/A064648. There is no special name for it. I haven't found anything about the second constant, either. – 2017-01-28
  • 4
    [See also](https://www.cambridge.org/core/journals/mathematical-gazette/article/div-classtitle9929-on-the-sum-of-the-reciprocals-of-the-primorialsdiv/56ECB260FAE52588FCECE9AAAAB2695D) – 2017-01-28
  • 1
    And the second number is here: https://oeis.org/A132120 – 2017-01-28
  • 0
    [both are irrational](http://math.stackexchange.com/a/1864796/276986) – 2017-01-28

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