If $a_{n} \not\to 0$ then $\sum_{n=1}^{\infty} a_{n}$ does not converge **Proof**: We will show that if $\sum_{n=1}^{\infty}a_{n}$ converges then $a_{n} \to 0$ as $n \to \infty$ Suppose that $\sum_{n=1}^{\infty}a_{n} = A$. I.e. $\lim_{ k \to \infty } \sum_{n=1}^{k}a_{n} \to A$By [[Shift Rule for Limits|shift rule]], $\lim_{ k-1 \to \infty } \sum_{n=1}^{k-1} a_{n} \to A $So $a_{k} = \sum_{n=1}^{k} a_{n} - \sum_{n=1}^{k-1}a_{n} \to A-A = 0 \text{ as } k \to \infty $ **Note**: $a_{n} \to \infty$ does not imply $\sum a_{n}$ converges. For example the [[Harmonic Numbers]] does not converge. See application [[Ratio Test for Series]].