Definitions:
Given a sequence $\{a_n\}$, define $$s_n= \sum_{j=0}^n a_j.$$ The sequence $\{s_n\}$ is called the series of partial sums of $\{a_n\}$. A series is convergent if $\{s_n\}$ has finite limit; devergent if $\{s_n\}$ has an infinite limit; and indeterminate if $\{s_n\}$ has no limit.
Known facts:
We have some criteria to determine convergence or divergence of a series if $\{s_n\}$ are all positive (or all negative) for $n$ greater than a certain index $n_0$ (ratio, root, integral tests, etc) or if $\{s_n\}$ has alterning signs (Liebnitz test).
Problems and questions:
Suppose that a series $ \sum a$ has not all non-negative (or non-positive) terms. Suppose also that we know (for example, by seeing that $a_n \to l \neq 0$) that it is not-convergent; what are some strategies to deduce if the series is divergent or indeterminate?
In particular, is it true that if $a_n \to l \in \mathbb{R} \cup \{\pm \infty\}$ then the series diverge and if $\lim_n a_n$ doesn't exist then the series is indeterminate? Why?