I have a question about the St-Petersburg paradox. In the case of expected utility with log utility function, how can we show analytically that, for $w > 2$, $c$ (limit price) is increasing in $w$ (initial wealth) where $c$ is implicitly defined by the equation
$ \sum_{k=1}^{\infty} \frac{1}{2^{k}} [\ln(w+2^{k-1}-c) - \ln(w)] = 0$?
I saw a few answers with numerical solutions, but I would like to know whether there is a way to show this analytically. If I use the above expression to obtain $dc/dw$, I get
$\frac{dc}{dw} = 1 - \frac{ \frac{1}{w} } {\sum_{k=1}^{\infty} \frac{1}{2^{k}} \frac{1}{w+2^{k-1} - c}}$
where $c$ is defined by the first expression and thus a function of $w$.
From the numerical solutions I have seen, I think $dc/dw>0$, but how do we show it analytically?
Thanks!