Convergent series terms go to zero

Corollary: If $\sum_{n=1}^\infty a_n$ converges in $\mathbb{R}$ or $\mathbb{C}$, then $a_n\to 0 \quad\text{as } n\to\infty.$

This is a necessary condition for convergence of a series (but far from sufficient).

Connection to parent theorem: Let $s_N=\sum_{n=1}^N a_n$ be the partial sums . If $s_N\to s$, then $a_N = s_N - s_{N-1}\to s-s=0.$