Problem
of the
Week
PROBLEM 213

This problem follows up a sequence introduced in POTW 187.  It was suggested by Lou Cairoli.

Define a sequence of natural numbers p0, p1, p2,... as follows: 

Put p0 = 2, and for k > 0 define pk+1 = p0p1×××pk+ 1; 

in words, the next term in the sequence is one more than the product of all the previous terms.  The sequence begins

p0 = 2
p1 = 3 = 2 + 1
p2 = 7 = 2×3 + 1
p3 = 43 = 2×3×7 + 1
p4 = 1807 = 2×3×7×43 + 1
p5 = 3263443 = 2×3×7×43×1807 + 1
p6 = 10650056950807 = 2×3×7×43×1807×3263443 + 1

Consider the sum of the reciprocals of the sequence p0, p1, p2, p3,...:

Does this series converge?  If so, what is its sum?

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ã2005 Alberto L. Delgado