Convergence properties of related series

Question

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.

Answer 1

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 $u_n$.