For better clarity, I will report the definitions we are using: given a probability space $(\Omega, \mathcal{F}, P)$ and a family $(X_n)_{n=1}^\infty $ of independent random variables $\Omega \longrightarrow \mathbb{R}$, we define the $\sigma$-algebras: $$\mathcal{F}^n := \sigma (X_n, X_{n+1}, \dots ) \qquad \qquad \mathcal{F}^\infty = \bigcap _{n=1}^\infty \mathcal{F}^n $$
Clearly an event (or a random variable) is $\mathcal{F}^\infty$-measurable if and only if is $\mathcal{F}^n$-measurable for every $n \in \mathbb{N}$.
We want to see that the event $$\left\{ \sum_{n=1}^\infty X_n \le 1 \right\} $$ is not $\mathcal{F}^\infty$-measurable, while the event $$\left\{ \sum_{n=1}^\infty X_n < +\infty \right\} $$ is $\mathcal{F}^\infty$-measurable.
Heuristically, it is clear to me that an event is $\mathcal{F}^\infty$-measurable if we can "remove" the first terms without changing nothing, so for example if the series $\sum _{n \ge1} X_n$ converges, then also the series $\sum _{n \ge n_0} X_n$ converges, for every $n_0 \in \mathbb{N}$. Viceversa, we can have that $\sum _{n \ge 1}X_n >1$ and $\sum _{n \ge n_0}X_n \le 1$ for some $n_0 \in \mathbb{N}$.
The point is that these are just heuristical observations, but I want to demonstrate these two facts in a mathematically rigorous manner. For example, I was trying to prove that the second event is $\mathcal{F}^n$-measurable for every $n \in \mathbb{N}$ and that the first event is not (which means that exists at least a $n_0 \in \mathbb{N}$ such that the first event is not $\mathcal{F}^{n_0}$-measurable), but I can't figure out how to do it.