A sequence of real numbers {B(m)} is such that:

B(1) = 1, B(2) = 2, and:

B(m+1) = 1 + B(1)*B(2)*…..*B(m), whenever m ≥ 2

Evaluate:

Limit  (1/B(1) + 1/B(2) + ……+ 1/B(t))
t → ∞

The recipricol values of terms of the series are:
(1, 2, 3, 7, 43, 1807, 3263443, 10650056950807,...).

The sequence (2, 3, 7, 43, 1807, 3263443, 10650056950807,...) are the values of Sylvester's sequence, which each member of the sequence is the product of the previous plus one. It is known that the sum of the reciprocals of Sylvester's sequence converges to 1.

With adding the additional term, 1, to the sum, the series  converges to 2.

Edited on February 23, 2008, 6:50 am

Posted by Dej Mar on 2008-02-22 11:49:36