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**Need help with euler,s transform of series....**

I want to summ the series of general term:

(a(n)+1)Z**n from n 0 to infinite.. where a(n) is

a(n) 1 iff n is prime, and 0 elsewhere.

I have tried to use an euler,s transform to improve convergence..but is this trick legal for this series?..

In fact as you will have know the series is related to the sum of all primes for the function z**(x) where z would be a number...

i need to know if i can use euler,s tranform into my series or another trick to improve convergence.