If $a_1 > 0$ and $$a_{n+1} = \ln\left[\frac{\exp(a_n)-1}{a_n}\right]$$ then what is the value of $$a_1 + a_1 a_2 + a_1 a_2 a_3 + a_1 a_2 a_3 a_4 + \cdots?$$
I have proved that this sequence is a decreasing one and converging to zero, but can't figure out what to do next. Any leads appreciated.
Writing $b_n = e^{a_n} - 1$, we have $b_{n + 1} + 1 = b_n / a_n$, or $b_n = a_n + a_n b_{n + 1}$.
By induction, we have $b_1 = a_1 + a_1a_2 + \dotsc + a_1a_2\dotsc a_n + a_1a_2\dotsc a_nb_{n + 1}$ for any $n \geq 1$.
Letting $n$ tends to infinity, and noting that $\lim\limits_{n\rightarrow \infty} a_n = \lim\limits_{n\rightarrow \infty}b_n = 0$, we see that the infinite sum converges to $b_1 = e^{a_1} - 1$.