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Convergent series terms go to zero

If a series converges, its terms must converge to 0

Corollary: If ∑n=1∞an\sum_{n=1}^\infty a_n in R\mathbb{R} or C\mathbb{C}, then

an→0as n→∞.a_n\to 0 \quad\text{as } n\to\infty.
Remarks

Connection to parent theorem: Let sN=∑n=1Nans_N=\sum_{n=1}^N a_n be the . If sN→ss_N\to s, then

aN=sN−sN−1→s−s=0.a_N = s_N - s_{N-1}\to s-s=0.

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