I am considering a stochastic process (say waiting for an Uber) with arrival time $T$, where $T=\frac1U$ with $U\sim\textrm{Unif}(0,1)$. I am willing to wait for a constant time $\tau<T$ before calling for another Uber. I am told that I should be minimising $\boxed{\frac{1-\ln\tau}{1-\tau}}$ to minimise the worst case waiting time (or what I think is the worst case waiting time). Why is this so? Here's what I've done so far:
i) $T$ has the inverse uniform distribution with support $(1,\infty)$ and PDF $\frac1{t^2}$ and CDF $1-\frac1{t^2}$. (Attempting to) integrate shows that $\mathbb{E}(\frac1X)$ diverges.
ii) The worst case waiting time is given by $W=\max\{T_1,T_2+\tau\}$ where $T_1$ is the waiting time for the first Uber, and $T_2$ is the waiting time for the second Uber after giving up on the first Uber. Then $$\mathbb{P}(W\leq w)=\mathbb{P}(T_1\leq w)\mathbb{P}(T_2+\tau\leq w)=\mathbb{P}(T_1\leq w)\mathbb{P}(T_2\leq w-\tau)=\int_1^w\int_1^{w-\tau}\frac1{u^2}\frac1{v^2}\textrm{d}u\textrm{d}v=\left(1-\frac1{w^2}\right)\left[1-\frac1{(w-\tau)^2}\right],$$ which means that $$\mathbb{E}(W)=\int_1^\infty w\left(1-\frac1{w^2}\right)\left[1-\frac1{(w-\tau)^2}\right]\textrm{d}w.$$
However, the integral I obtained diverges, and I have no idea how to "massage" my reasoning to obtain the required answer. Where have I gone wrong/is my interpretation of the problem correct?