Expected number of frog jumps

Let $X_i$ denote the distance of jump $i$ of the frog. Each $X_i$ is independent of one-another. The total distance at jump $n$ is then $D_n=X_1+\cdots+X_n$. Define $\tau$ for $D_\tau$, where $\tau$ is the first time such that $D_\tau\geq d$. You are interested in $E(\tau)$. The distribution of $D_n$ is the $n$-fold convolution of $F$'s: $F_n = (F\star)^n:=F\star F\star\cdots \star F$.

$P(\tau> n) = P(D_n<d)$

Now use $E(\tau)=\sum_{n=0}^\infty P(\tau> n)=\sum_{n=0}^\infty P(D_n<d)$

Which we can simplify when $F$ is continuous by noting $P(D_n<d)=P(D_n\leq d)=F_n(d)$. I'm not sure if there's any more simplification you can do without an explicit form for $F$.

Reference: This problem has been worked out thoroughly if the random walk steps are uniform and continuous. See, for example,

Russell, K. G. On the number of uniform random variables which must be added to exceed a given level. J. Appl. Probab. 20 (1983), no. 1, 172–177.

It is a well-known puzzle to show that the expected number of uniform(0,1) steps to exceed level 1 is $\exp(1)$. I was sure that this appears somewhere on this site, but I can't find it.