[seqfan] Help with a calculation for A001097 (twin primes)

[Bob Selcoe]

Hello Seqfans,

Is there a way to calculate the following limit as n approaches infinity?  This will be helpful in a comment I intend to add to A001097 (twin primes)

Not sure about the most efficient or best way to notate, so here's what I'm asking:

For a(n)>1, let a(n) = A007310 (numbers congruent to 1 or 5 mod 6): 5, 7, 11, 13, 17, 19, 23, 25.... 

So a(1) = 5, a(2) = 7, a(3) = 11 etc.

Calculate 1/a(1)  +  1/a(2)*(1 - 1/a(1))  +  1/a(3)*(1 - 1/a(1) - 1/a(2))  +  1/a(4)*(1 - 1/a(1) - 1/a(2) - 1/a(3))...

So 1/5 + 4/35 + 23/385 + 218/5005 +...  

If this is not easily solved, could someone please run a program to get an approximation?  

Thanks in advance!

 Cheers,
Bob Selcoe