Tools for determining convergence of non-specific sums. I've done a lot of googling and searching on here and I can't find what I'm looking for. So at risk of asking a duplicate question, I'm looking for some tools or methods to figure out if a sum is convergent or divergent. The sum in question is
$$\sum_{i=1}^{\infty} \frac{A_{i}}{\prod_{j=1}^{i} B_{j}} = \frac{A_1}{B_1} + \frac{A_2}{B_{1}B_{2}} + \frac{A_3}{B_{1}B_{2}B_{3}} + \dots$$ Where $A_i$, $B_i$ are integers $\geq 1$ for all $i$. I don't know what any value of $A_i$ or $B_i$ is specifically, and I don't really care what it might sum to, I just need to know if it converges or not.
That said, are there any tools or methods that anyone might recommend to help me figure this out?
 A: I think actually the ratio test might be helpful: It says that a series $\sum_{i=1}^\infty a_i$ is absolutely convergent (i.e. the sum of all $|a_i|$ is convergent) if $a_i\neq 0$ for all $i$ and there is a $c < 1$ and $N\in \mathbb{N}$ so that
$$
\left| \frac{a_{i+1}}{a_i} \right|\leq c \quad\forall i\geq N
$$
In this case we can calculate
$$
\left| \frac{a_{i+1}}{a_i} \right| = \frac{\frac{A_{i+1}}{\prod_{j=1}^{i+1}B_j}}{\frac{A_i}{\prod_{j=1}^{i}B_j}}= \left|\frac{A_{i+1}}{A_i\cdot B_{i+1}}\right| \overset{!}{\leq} c
$$
So you get the following criterion: If you can find an index $N\in\mathbb{N}$ so that for all $i\geq N$ the condition
$$
\left|\frac{A_{i+1}}{A_i\cdot B_{i+1}}\right| \leq c
$$
is fulfilled for a $c<1$, the series is (absolute) convergent.
Examples
As mentioned in my comment above you can verify those two examples with the above criterion:

*

*$A_i=a, B_i=10^{-i}$: In that case we have
$$
\frac{A_{i+1}}{A_i\cdot B_{i+1}} = \frac{1}{1\cdot 10^{-i-1}} = 10^{i+1} > 1
$$
for all $i > 0$, so the series is divergent.


*$A_i=1, B_i=2$: We then have $\frac{1}{1\cdot 2} = \frac 1 2 \leq c=\frac 1 2$, so the series is convergent.
Addition for negative values
You might get wrong results when including negative values. As the above criterion only verifies absolute convergence (which is stronger than convergence), you might get series that are not absolute convergent but convergent which will be not detected by this formula.
One example would be $A_i=1,B_i=-\frac{i+1}{i}$: This results in the alternating harmonic series which is convergent but not absolutely convergent. Indeed, the above test results into
$$
\left|\frac{1}{1\cdot\left(-\frac{i+2}{i+1}\right) }\right| = \frac{i+1}{i+2}
$$
and there exists no number $c<1$ so that $\frac{i+1}{i+2} \leq c$ for all $i>N$ because this fraction actually approaches $1$. This verifies that the series is not absolutely convergent (but says nothing about the convergence).
So you might find another criterion when handling negative values.
