given-the-recursive-a-n-define-by-setting-a-1-0-1-a-n-1-a-n-1-a-n-n-1-prove-that-1-lim-n-na-n-1-2-b-n-n-1-na-n-is-a-incresing-sequence-and

Question Number 216785 by universe last updated on 20/Feb/25

given the recursive {a_{n}} define by setting

a_{1} \in (0, 1), a_{n + 1} = a_{n} (1 - a_{n}), n ⩾ 1

prove that (1) \lim_{n \to \infty} {na}_{n} = 1

(2) b_{n} = n (1 - {na}_{n}) is a incresing sequence

and diverge to \infty

(3) \lim_{n \to \infty} \frac{n (1 - {na}_{n})}{\ln (n)} = 1