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COMMENT
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By re-writing the sequence of sums as 1-Product[1-1/Prime[n]], one can show that the product goes to zero and the sequence of sums converges to 1. This is interesting because the terms approach 1/(2*Prime[n]) for large n, and a sum of such terms might be expected to diverge, since Sum{1/(2*Prime[n]} diverges.
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MATHEMATICA
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Table [1- Product[1 - (1/Prime[k])), {i, 1, j}, {j, 1, 20}]; This is a table of the individual sums: Sum[Product[ 1 - (1/Prime[k]), {k, n-1}]/Prime[n], {n, 1, 3}], which is the sum of terms of the Mathematica table given in A038111 (three terms, in this example).
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