Probability of finding the fly inside a range A fly is traversing the non-negative x-axis. It starts at $x_0=k$. At the $i^{th}$ step (starting from the zeroth step), it uniformly randomly jumps to a point in the range $[0,x_i]$.
Probability that the fly is in the range $[a,b]$ after $n$ jumps is denoted by $P_n(a,b)$. Find
$$\lim_{y \to 0}\frac{P_n(1,1+y)}{y}$$
Any help will be appreciated.
EDIT:
Sorry for the late response. Here is my brief try. If we consider the condition that after $n$ jumps, the fly is at a position $> a$, but exclude the condition that it should be $< b$. Let $p=a/k$
\begin{align}
\begin{split}
P_n(a)&=\int\limits_{x1=a}^k ~\int\limits_{x_2=a}^{x_!}....\int\limits_{x_n=a}^{x_{n-1}}\frac{1}{k}\frac{1}{x_1}...\frac{1}{x_{n-1}}\ dx_1 \ dx_2 ... \ dx_n\\
&=1-p+p\sum_{i=1}^{n-1}\frac{(\log p)^i}{i!}
\end{split}
\end{align}

EDIT-2: Let $p=a/k$ and $q=b/k$
\begin{align}
\begin{split}
P_n(a,b)&=\int\limits_{x1=a}^k ~\int\limits_{x_2=a}^{x_!}....\int\limits_{x_{n-1}=a}^{x_{n-2}}\int\limits_{x_n=a}^{\min (x_{n-1},b)}\frac{1}{k}\frac{1}{x_1}...\frac{1}{x_{n-1}}\ dx_1 \ dx_2 ... \ dx_n \\
&=q-p+\sum_{i=1}^{n-1}\frac{(-1)^i}{i!}\left[q(\log q)^i-p(\log p)^i\right]
\end{split}
\end{align}
Substituting $a=1$ and $b=1+y$, and evaluating the limit,
\begin{align}
\begin{split}
\lim_{y \to 0}\frac{P_n(1,1+y)}{y}=\frac{(-1)^{n+1}}{(n-1)!k}\left(\log \frac{1}{k}\right)^{n-1}
\end{split}
\end{align}
