I've made this question some time ago. I thought that the term $x$ in $x+e^x$ would be a sequence $a_x=x=\{0,1,2,3,\ldots\}$ and it turns out that it is the sequence $\{0,1,0,0,\ldots\}$.
Whenever I see an exercise now, I assume that $x$ has the sequence I pointed, and when I see $x^2$, I assume the sequence as $\{0,0,1,0,\ldots\}$. The problem is that I'm not really sure why that is so, I just saw some examples and copied without really understanding it. My guess is that it has something to do with this and this.
Does the summation method pointed in these comments indicate a way of discovering it? I am having a introduction to combinatorial analysis and we didn't have a thorough training in summations, nor training in series (I'm currently reading Knuth/Graham/Patashnik to suplement this). But I'd like to have a brief direction of how to discover this.
To determine if a sequence goes on forever, you can use ellipses such as in {$\cdots-2, -1, 0, 1, 2\cdots$} showing that they do that, like you did. It can also be identified by when something says "all integers", "all even numbers greater than $-5$", "all numbers divisible by $10$", etc. The word "all" identifies that any of those sequences goes on forever meaning it doesn't have a finite number of numbers in it just for the ones I showed you and ones that have the same thing.