Which series converges question

Question

Which of the following statements are true. Given $S_1, S_2$, where

$S_1:$ A series $$\sum_{n=0}^{\infty}a_n$$ converges if for a given $\epsilon\gt0$ there exists $N_o \in N$ such that $|a_{n+1}-a_{n}|\lt \epsilon$ for all $n\ge N_o$.

$S_2:$ A series $$\sum_{n=0}^{\infty}a_n$$ converges if $|a_{n+1}-a_{n}|\lt \alpha^n$ where $\alpha$ is a fixed real no in $(0,1)$.

Answer 1

Consider the case of $a_n=1$ and notice that the series don't converge in this case for both scenarios as there has to be the condition of $a_n$ converging to zero as a separate condition here.