= Solution
The first event is a <tail event>: for any $m$ it equals $\bigcup_{N\geq m}\bigcap_{n\geq N}\{X_n\leq0\}$, measurable in $\sigma(X_m,X_{m+1},\ldots)$.
Convergence of the partial sums is also a <tail event>. For any $m$, subtracting the finite real sum $X_1+\cdots+X_{m-1}$ shows that convergence is equivalent to convergence of $\sum_{j=m}^nX_j$. The Cauchy criterion expresses the latter using countable unions and intersections of inequalities involving these tail sums, so it belongs to every <tail sigma-algebra>.
The event that the partial sums are nonpositive infinitely often is \b[not generally a <tail event>]. For a counterexample let $X_1$ take values $-1$ and one on a two-point <probability> space, each with positive <probability>, and let $X_n=0$ for $n\geq2$. The event is then $\{X_1=-1\}$, while $\sigma(X_2,X_3,\ldots)$ is trivial. The variables in this example are even independent. Its status can be different for particular sequences, but it is not tail-measurable in general.
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