Let $u(b)$ be a uniformly random number bounded by $0\le u(b)\le b$ and let $v(n)=\underbrace{u\circ u\circ...\circ u}_{n\text{ times}}\phantom{ }\circ u(1)$. What on average is $\sum\limits_{n=0}^\infty v(n)$?
I created a graph on desmos to test. I conjecture that it is $2$ because after $1691$ iterations, the mean was $2.00249000071$. How can this be shown analytically?
EDIT: My only idea of how to approach this problem is to get the mean value $\mu$ of a uniformly sampled value between $0$ and $n$. For $n=1$, $\mu=\frac{1}{2}$. I can see that if I keep going, I will get $\mu=\frac{n}{2}$ which means the sum is equal to $\sum\limits^{\infty}_{n=0}2^{-n}=2$. Is this thinking right?