On an entry of Ramanujan in his Notebooks: a nested roots expansion

In this letter, the elementary result of Ramanujan for nested roots, also called continued or infinite radicals, for a given integer N, expressed by him as a simple sum of three parts (N=x+n+a) is shown to give rise to two distinguishably different expansion formulas. One of these is due to Ramanujan and surprisingly, it is this other formula, not given by Ramanujan, which is more rapidly convergent!