As a follow-up to the question Does $\lim\limits_{n\to\infty}\int\limits_{0}^{1}\{x+\frac{1}{2}\{x+\frac{1}{3}\{x+...\frac{1}{n}\}\}\}dx$ converge? where it was shown that $\lim\limits_{n\to\infty}\int\limits_{0}^{1}\{x+a_1\{x+a_2\{x+...a_n\}\}\}dx$ does converge for the sequence $a_n=\frac{1}{n}$, I ask for something more generalized: For what sequences $a_n$ does the integral converge for? Must $\lim\limits_{n\to\infty}a_n$ exist? What tests can we apply to a sequence to find whether it converges?
I will use:
$$I_n\left(a_n\right)=\lim\limits_{n\to\infty}\int\limits_{0}^{1}\{x+a_1\{x+a_2\{x+...a_n\}\}\}dx$$
I know through observation that for all integer sequences $a_n$, $I_\infty(a_n)=\frac{1}{2}$.
