Theorem: If $\lim_{n\to\infty}a_n\neq 0$ then the series $\sum_{n=1}^{\infty}a_n$ diverges

Proof

$$a_n=S_n\mathbin{-}S_{n\mathbin{-}1}$$

$$\lim_{n\to\infty}a_n=\lim_{n\to\infty}S_n\mathbin{-}\lim_{n\to\infty}S_{n\mathbin{-}1}\neq0$$

$$ \lim_{n\to\infty}S_n \neq \lim_{n\to\infty}S_{n \mathbin{-}1}$$

Thus $S_n$ does not tend to a definite limit as $n$ tends to infinty. This means no limit exists and the series diverges.

On the other hand no conclusions can be drawn if

$$\lim_{n\to\infty}a_n = 0$$

All we know is that the terms tend to zero. The series may converge if the terms tend to zero rapidly enough as in the case of $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$. Else the series may diverge if the terms tend to zero slowly as in the case of $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$.