Does $a_n$ converge if $a_{n+1} = a_n + \frac{1}{e^{a_n}+1}$? Or does its convergence depend on $a_0$?

Question

Does $a_n$ converge if$a_{n+1} = a_n + \frac{1}{e^{a_n}+1}$? Or does its convergence depend on $a_0$? As described in the title, it seems intuitively that it should converge, but I don't know how to prove it.

Answer 1

$a_n$ is nondecreasing. If $a_n$ converges then the series $a_{n+1} -a_n \to 0$ converges and so $\lim\limits_{n\to \infty} \frac1{e^{a_n} + 1} =0$ so $\frac1{e^{\ell} + 1} = 0$ which is impossible. This implies that $\lim\limits_{n\to\infty} a_n = \infty$