$\newcommand{\ga}{\gamma} $ Of course, without further assumptions on the $e_t$'s, no good bound can be given. However, looking at your examples, it appears that you are primarily interested in situations where the $e_t$'s satisfy the following conditions: For some real constant $c\ge1$ and some positive log-convex real sequence $(f_t)$ we have the following: (i) $f_t\le e_t\le c f_t$ for all natural $t$ and (ii) $f_{t+1}/f_t\to1$ as $t\to\infty$, so that $\rho_N:=(f_N/f_1)^{1/(N-1)}\to1$ as $N\to\infty$. In fact, in all your examples except for $e_t=(\ln t)/t$, we can use $c=1$ and $f_t=e_t$ for all $t$.

So, for all $N$ large enough for the inequality $\ga<\rho_N$ to hold, we have \begin{equation} \sum_{t=1}^N\ga^t e_{N-t}\le c\sum_{t=1}^N\ga^t f_{N-t}\le c\sum_{t=1}^N \ga^t f_N^{1-t/(n-1)}f_1^{t/(n-1)} =cf_N\sum_{t=1}^N(\ga/\rho_N)^t\le cf_N\sum_{t=0}^\infty(\ga/\rho_N)^t =\frac{cf_N}{1-\ga/\rho_N}\lesssim\frac{ce_N}{1-\ga}, \end{equation} as you observed empirically.

One can do similarly assuming (instead of the above conditions involving the $f_t$'s) that the sequence $(e_t)_{t=t_0}^\infty$ is log convex for some natural $t_0$ and $e_{t+1}/e_t\to1$ as $t\to\infty$. In all your examples we can take $t_0=1$ -- except for $e_t=(\ln t)/t$, where we can take $t_0=5$.

$\newcommand{\ga}{\gamma} $ Of course, without further assumptions on the $e_t$'s, no good bound can be given. However, looking at your examples, it appears that you are primarily interested in situations where the $e_t$'s satisfy the following conditions: For some real constant $c\ge1$ and some positive log-convex real sequence $(f_t)$ we have the following: (i) $f_t\le e_t\le c f_t$ for all natural $t$ and (ii) $f_{t+1}/f_t\to1$ as $t\to\infty$, so that $\rho_N:=(f_N/f_1)^{1/(N-1)}\to1$ as $N\to\infty$. In fact, in all your examples except for $e_t=(\ln t)/t$, we can use $c=1$ and $f_t=e_t$ for all $t$.

So, for all $N$ large enough for the inequality $\ga<\rho_N$ to hold, we have \begin{equation} \sum_{t=1}^N\ga^t e_{N-t}\le c\sum_{t=1}^N\ga^t f_{N-t}\le c\sum_{t=1}^N \ga^t f_N^{1-t/(N-1)}f_1^{t/(N-1)} =cf_N\sum_{t=1}^N(\ga/\rho_N)^t\le cf_N\sum_{t=0}^\infty(\ga/\rho_N)^t =\frac{cf_N}{1-\ga/\rho_N}\lesssim\frac{ce_N}{1-\ga}, \end{equation} as you observed empirically.

One can do similarly assuming (instead of the above conditions involving the $f_t$'s) that the sequence $(e_t)_{t=t_0}^\infty$ is log convex for some natural $t_0$ and $e_{t+1}/e_t\to1$ as $t\to\infty$. In all your examples we can take $t_0=1$ -- except for $e_t=(\ln t)/t$, where we can take $t_0=5$.

$\newcommand{\ga}{\gamma} $ Of course, without further assumptions on the $e_t$'s, no good bound can be given. However, looking at your examples, it appears that you are primarily interested in situations where the $e_t$'s satisfy the following conditions: For some real constant $c\ge1$ and some positive log-convex real sequence $(f_t)$ we have the following: (i) $f_t\le e_t\le c f_t$ for all natural $t$ and (ii) $f_{t+1}/f_t\to1$ as $t\to\infty$, so that $\rho_N:=(f_N/f_1)^{1/(N-1)}\to1$ as $N\to\infty$. In fact, in all your examples except for $e_t=(\ln t)/t$, we can use $c=1$ and $f_t=e_t$ for all $t$.

So, for all $N$ large enough for the inequality $\ga<\rho_N$ to hold, we have \begin{equation} \sum_{t=1}^N\ga^t e_{N-t}\le c\sum_{t=1}^N\ga^t f_{N-t}\le c\sum_{t=1}^N \ga^t f_N^{1-t/(n-1)}f_1^{t/(n-1)} =cf_N\sum_{t=1}^N(\ga/\rho_N)^t\le cf_N\sum_{t=0}^\infty(\ga/\rho_N)^t =\frac{cf_N}{1-\ga/\rho_N}\sim\frac{ce_N}{1-\ga}, \end{equation} as you observed empirically.

$\newcommand{\ga}{\gamma} $ Of course, without further assumptions on the $e_t$'s, no good bound can be given. However, looking at your examples, it appears that you are primarily interested in situations where the $e_t$'s satisfy the following conditions: For some real constant $c\ge1$ and some positive log-convex real sequence $(f_t)$ we have the following: (i) $f_t\le e_t\le c f_t$ for all natural $t$ and (ii) $f_{t+1}/f_t\to1$ as $t\to\infty$, so that $\rho_N:=(f_N/f_1)^{1/(N-1)}\to1$ as $N\to\infty$. In fact, in all your examples except for $e_t=(\ln t)/t$, we can use $c=1$ and $f_t=e_t$ for all $t$.

So, for all $N$ large enough for the inequality $\ga<\rho_N$ to hold, we have \begin{equation} \sum_{t=1}^N\ga^t e_{N-t}\le c\sum_{t=1}^N\ga^t f_{N-t}\le c\sum_{t=1}^N \ga^t f_N^{1-t/(n-1)}f_1^{t/(n-1)} =cf_N\sum_{t=1}^N(\ga/\rho_N)^t\le cf_N\sum_{t=0}^\infty(\ga/\rho_N)^t =\frac{cf_N}{1-\ga/\rho_N}\lesssim\frac{ce_N}{1-\ga}, \end{equation} as you observed empirically.

One can do similarly assuming (instead of the above conditions involving the $f_t$'s) that the sequence $(e_t)_{t=t_0}^\infty$ is log convex for some natural $t_0$ and $e_{t+1}/e_t\to1$ as $t\to\infty$. In all your examples we can take $t_0=1$ -- except for $e_t=(\ln t)/t$, where we can take $t_0=5$.