I suspect that there is a mistake in the Wikipedia article on the St Petersburg paradox, and I would like to see if I am right before modifying the article.
In the section "Solving the paradox", the formula for computing of the expected utility of the lottery for a log utility function is given to be
$$E(U)=\sum_{k=1}^\infty \frac{(\ln(w+2^{k-1}-c)-\ln(w))}{2^k} <\infty$$
I do not see why the term $\ln(w)$ should be inside the summation (neither why it should be divided by $2^k$ by the way). Do you see anything I overlooked which would justify this?
Related question : Maximum amount willing to gamble given utility function $U(W)=\ln(W)$ and $W=1000000$ in the game referred to in St. Petersberg's Paradox?
Presumably you mean that the formula should be:
$$E(U)=\left(\sum_{k=1}^{\infty} \frac{\ln(w+2^{k-1}-c)}{2^k}\right)-\ln(w)$$
But since $\sum_{k=1}^\infty \frac{1}{2^k} = 1$, this is identical to the formula from the Wikipedia page.