Consider the following sum s(k), of k terms:
s(k) = sum for i = 1 to k of 1/((kW/(1-W)) + i -1)
where k and i are integers, and W is the Omega constant defined by ln(1/W) = W (W = 0.567143)
As an example, for k = 3, we have:
s(3) = 1/(3W/(1-W)) + 1/((3W/(1-W)) +1) + 1/((3W/(1-W)) +2)
Numerical analysis shows that the limit of this sum as k tends to infinity is W (the Omega constant). I am interested in finding a step-by-step analytical proof that this is the case.
A nice proof, or simply useful suggestions on how to attack the problem would be much appreciated.