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 → ∞

Write S(M) = 1/B(1) + ... + 1/B(M), then

S(2) = 2 - 1/2,  so S(3) = 2 - 1/(2*3) = 2 - 1/[ B(1) *B(2) *B(3)] = 2 - 1/[B(4) -1]

By induction we can see that S(M) = 2 - 1/[B(M+1) - 1]

clearly B(M) goes to infinity, so S(M) goes to 2

Posted by FrankM on 2008-02-23 07:46:41