Answer
False.
Work Step by Step
The statement is false.
Counter example:
Consider $a_n=\frac{1}{n}$. We have $\lim\limits_{n \to \infty}a_n=0$, but $\sum_{n=1}^{\infty}a_n=\sum_{n=1}^{\infty}\frac{1}{n}$ does not converge.
Therefore, if $\lim\limits_{n \to \infty}a_n=0$, then $\sum_{n=1}^{\infty}a_n$ may or may not converge.
