Let \(\sum a_n\) be convergent series and let \(s_n = a_1 + a_2 + a_3 + \cdots + a_n\). This means \(a_n = s_n - s_{n-1}\). If \(\sum a_n\) is convergent, then the sequence \(\{s_n\}\) is convergent. Let \(\lim_{n \to \infty} s_n = s\). If \(n-1 \to \infty\) when \(n \to \infty\), then \(\lim_{n \to \infty} s_{n-1} = s\). Therefore:
\[\begin{align} \lim_{n \to \infty} a_n &= \lim_{n \to \infty} (s_n - s_{n-1}) \\ &= \lim_{n \to \infty} s_n - \lim_{n \to \infty} s_{n-1} = s - s = 0\end{align}\]
This shows if \(\sum a_n\) is convergent, then \(\lim_{n \to \infty} a_n = 0\). The contrapositive of this is "If \(\lim_{n \to \infty} a_n \ne 0\) or \(\lim_{n \to \infty} a_n\) does not exist, then \(\sum a_n\) is divergent".