Convergence properties of related series

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

$$sumlimits_{n in mathbb{N} } g_n = infty,
(1)$$

$$sumlimits _{m in mathbb{N}} u_m expleft{ – sumlimits _{i = 1} ^m g _i u_i right } = infty?
(2)$$

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

$$
sumlimits _{m in mathbb{N}} ln m expleft{ – sumlimits _{i = 1} ^m frac 1i right } approx
sumlimits _{m in mathbb{N}} ln m expleft{ – ln m right }
=sumlimits _{m in mathbb{N}} frac{ln m }{m} = infty.
$$

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.