I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function?
I tried to express $U_{n}$ as a function of $n$, I tried expressing it as a function of $U_{n-1}$, I tried looking at $U_{n+1} - U_{n}$, all without success. I built an Excel macro to look at what the sequence looks like. With that, I can confirm that the sequence does have a limit (and different from zero, but depending from $a$ and $b$), after having tried several values for $a$ and $b$. I tried inferring the value of the limit from the Excel calculations, but it is not obvious.
Let $a$ and $b$ be natural numbers, with $1\leq a< b$
We define $U_{n}$ by:
$$ U_n := \begin{cases} \displaystyle \frac{1}{b+n} \left [ \frac{1}{a+n} + \sum_{i=0}^{n-1}\left ( U_{i} \sum_{k=n-i}^{b+n-1} \frac{1}{k} \right ) \right ] & \text{for $n\geq 1$,} \\[1ex] \displaystyle \frac{1}{ab} & \text{for $n=0$.} \end{cases} $$
I want to find $\lim_{n \to +\infty}U_{n}$.