Let $u\_m = \ln ^2 m$.
Does there exist a non-increasing sequence of positive numbers $\{g\_n\}\_{n \in \mathbb{N}}$, $g\_n \to 0$, such that


$$\sum\limits\_{n \in \mathbb{N} } g\_n = \infty, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$


$$\sum\limits \_{m \in \mathbb{N}} u\_m \exp\left\{ - \sum\limits \_{i = 1} ^m g \_i u\_i \right \} = \infty? \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ (2)$$


The answer would be positive if we took $u\_m$ to be $\ln m$, because in this case taking $g\_i = \frac{1}{i \ln i }$ would result in


$$
\sum\limits \_{m \in \mathbb{N}} \ln m \exp\left\{ - \sum\limits \_{i = 1} ^m \frac 1i \right \} \approx 
\sum\limits \_{m \in \mathbb{N}} \ln m \exp\left\{ - \ln m \right \}
=\sum\limits \_{m \in \mathbb{N}} \frac{\ln m }{m} = \infty.
$$


For (1) and (2) to hold simultaneously $\{g\_n\}\_{n \in \mathbb{N}}$ cannot converge to zero too quickly because (1) may fail, while converging too slowly may cause (2) to fail.


More generally, are there any results/techniques clarifying whether it is possible for two related series to have given convergence properties? In particular, is it possible to say something about a general case when $\{u\_n\}\_{n \in \mathbb{N}}$ is in a certain sense slowly increasing sequence diverging to $\infty$?


Any idea or reference would be kindly appreciated.



Convergence properties of related series
Call the two series $S\_1, S\_2$. Start out by letting $g\_1=1$. Whatever we do afterwards, this makes sure that $S\_1\ge 1$. Next, fix an $M$ such that $u\_M e^{-1\cdot u\_1}\ge 2$, and then give $g\_2, \ldots, g\_M$ a common small value that will give us
$$
e^{-\sum\_{j=2}^M g\_j u\_j}\ge \frac{1}{2} .
$$
This guarantees that $S\_2\ge 1$.


Now just continue in this way. Keep $g\_j$, $j>M$, constant for a while (if necessary) to make sure that $S\_1\ge 2$, and then turn your attention back to $S\_2$ etc.


Obviously, this procedure works for any unbounded sequence $u\_n$.


