TITLE: let $a_{n+1} = \log (1+a_n) $ for $n \ge 1$ . Check whether $\sum_{n=1}^{\infty} a_n$ converge or diverge?
QUESTION [5 upvotes]: let  $a_1$ be an arbitrary postive number and let $a_{n+1} = \log (1+a_n) $ for  $n \ge  1$.  Check  whether  $\sum_{n=1}^{\infty} a_n$  converges  or  diverges?
My attempt: I know  that $\log(1+x) = x - \frac{x^2}{2} + \frac {x^3}{3} \cdots$
Now  here  how can I conlcude  that  $\sum_{n=1}^{\infty} a_n$  converges or  diverges?
Any hints/solution will be appreciated.

REPLY [5 votes]: Look for hints within math.SE! Argue that $$\lim_{n\to\infty} n\cdot a_n=2$$ by referring to $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$. 
 Therefore the series $\sum a_n$ diverges by comparison to the harmonic series.