I'm interested in finding conditions about the convergence of a series. I have an increasing succession $\{a_n\}$.
The elements of $\{a_n\}$ are positive real numbers. I want to know the conditions where $$\sum_{n\gt1}^{\infty}\frac{1}{a_1a_2...a_n}$$ converge. I used the ratio test where i found that the series converges if $$\lim_{n \to \infty} \frac{1}{a_{n+1}}\lt1$$ Are there are more conditions?
I would appreciate additional insight, or hints.
If all of the $a_n$ are $\le 1$ then the $n$th term doesn't go to $0$ and the series diverges. Therefore, the condition you have found is necessary and sufficient.