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Following is an infinite series quite analogous to the exponential series, but composed of prime numbers only. $$\frac {1}{2} + \frac {1}{2.3} +\frac {1}{2.3.5} + \frac {1}{2.3.5.7} + \frac {1}{2.3.5.7.11} + …… ≈ 0.70523$$ How can we express it in closed form i.e. how can it be expressed as a combination of known constants (like $π, e,$ etc.) and/or special functions? The only pattern I can conceive is that the denominators form a part of Euclid's proof for infinite primes.

Awe Kumar Jha
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    The product of consecutive primes is called primorial numbers. Maybe this will be helpful? https://oeis.org/A064648 – TrostAft Oct 31 '18 at 04:13
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    Primorials grow pretty fast. So even if you do not have a closed form, you can get a pretty good idea of how many terms you need to compute this within machine epsilon. – parsiad Oct 31 '18 at 04:14
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    https://math.stackexchange.com/questions/2117676/whats-infinte-sum-of-the-reciprocal-of-the-primorial – parsiad Oct 31 '18 at 04:19
  • Although computational methods give exact numerical values yet a closed form is mathematically more meaningful as it tells a lot about the nature of the number, related properties and identities. – Awe Kumar Jha Oct 31 '18 at 04:34
  • I am 99.9999999999999% sure this sum wont have a closed form in terms of other known numbers and constant. – Nilotpal Sinha Oct 31 '18 at 05:29
  • As such it becomes more irresistible to define a new function that will characterise this constant number. – Awe Kumar Jha Oct 31 '18 at 05:41

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