shows that it diverges at $r_0$ for $S>2$ (it converges for $1<S<2$); from the integral (\[18122013a\]), we see that $w$ is infinite at $r_0$ for $S=2$.

### Choosing a value for $w_1$ {#23122013e}

Like $l_1$ and $l_2$, $w_1$ and $w_2$ will represent the places where the trip begins and ends, respectively. The ideal value for $w_1$ is the one that favors the conditions listed in Sec. \[29012014a\]. For reasons that will become clear in other sections (see, for instance, Sec. \[23122013a\]), we choose $r_1$ as $$r_1=M(1+S)^2/(2S^2). \label{29012014b}$$ One can easily check that this point is between $r_0$ and $r_m$ for $S>1$, which is the case we are interested in.

Defining $w_1$ as $-w(r_1)$ and using Eq. (\[18122013b\]), we get $$\begin{aligned}
w_1=M\biggl\{ (S^2-1)/(2S^2)+\ln\left(\frac{2S}{1+S} \right) +\sum_{n=2}^{\infty} \frac{[2S/(1+S)]^{n-1}-1}{(n-1)n!S(1+S)^{n-1}} \prod_{j=1}^{n}(S+2j-2) \biggr\}. \label{22122013a}\end{aligned}$$ From Fig. \[22012014a\], which is a plot of $w_1$ as a function of $S$ from $1$ to $\infty$, we see that $w_1$ is a monotonically increasing function of $S$ with its minimum at $S=1$ (note that $w_1=0$ for $S=1$). From Eq. (\[22122013a\]), one can evaluate the maximum value of $w_1$ (take the limit $S\to \infty$) and find that $0<w_1{ \put(2,0){$<$} \put(2,-5){$\sim$}\quad \ }2M$.

![This figure shows a plot of $w_1$ as a 