# Find limit of sequence defined by sum of previous terms and harmonics

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. I have been trying for days to find the limit of this sequence, I'm desperately hoping that someone could help find it. Any help is very much appreciated!

Let a and b be natural numbers, with $$1\leq a< b$$
We define $$U_{n}$$ by:

$$U_{n} = \frac{1}{b+n} * \left [ \frac{1}{a+n} + \sum_{i=0}^{n-1}\left ( U_{i} * \sum_{k=n-i}^{b+n-1} \frac{1}{k} \right ) \right ]$$ for $$n\geq 1$$

$$U_{0} = \frac{1}{a*b}$$

We want to find $$\lim_{n \to +\infty}U_{n}$$