convergence of a series $a_1 + a_1 a_2 + a_1 a_2 a_3 +\cdots$ Suppose all $a_n$ are real numbers and $\lim_{n\to\infty} a_n$ exists.
What is the condition for the convergence( or divergence ) of the series
$$ a_1 + a_1 a_2 + a_1 a_2 a_3 +\cdots $$
I can prove that $ \lim_{n\to\infty} |a_n| < 1 $ ( or > 1 ) guarantees
absolute convergence ( or divergence ).
What if $ \lim_{n\to\infty} a_n = 1 \text{ and } a_n < 1 \text{ for all } n $ ?

I believe it's indeterminant. Consider $a_n=1-\frac{1}{n+1}$ and $a_n=1-\frac{1}{\log_2{n+2}}$. Both converge to 1, but the first one converges very quicky - and ends up resulting in the harmonic series, and the second one does so very slowly, and I believe it does converge.
