Summary of my understanding of sequences and series' convergence and divergence? I'm trying to summarise my understanding of infinite sequences, series, and their relationships with respect to convergence at the fundamental level. Here is what I know. How much of this is correct?
First off, here's a table of the notations that I use, and their corresponding meaning.

My understanding is that:


*

*The sequence $\lbrace a_n \rbrace _{n=0}^{\infty}$ converges if
$$\lim\limits_{n\to\infty}a_n=L_{a}.$$

*The infinite series $\sum\limits_{n=0}^{\infty}a_n$ converges if its
sequence of partial sums,  $\lbrace s_n \rbrace _{n=0}^{\infty}$, has
a limit, i.e.$$\lim\limits_{n\to\infty}s_n=L_{s}.$$

*If the infinite series $\sum\limits_{n=0}^{\infty}a_n$ converges,
then the limit of the sequence $\lbrace a_n \rbrace _{n=0}^{\infty}$
is $0$, i.e. $$\sum\limits_{n=0}^{\infty}a_n \: converges \rightarrow
   \lim\limits_{n\to\infty}a_n=0.$$

*The divergence test: 
If the the limit of the sequence $\lbrace a_n \rbrace
   _{n=0}^{\infty}$ is NOT $0$ or does not exist, then the infinite series diverges, i.e. $$\lim\limits_{n\to\infty}a_n\neq0 \rightarrow \sum\limits_{n=0}^{\infty}a_n \: diverges$$
Would seriously appreciate it if anyone could verify whether the above is accurate or incorrect in any way.
EDIT: I've modified the two limits notation that were mentioned in the comments and answers below, as well as adding the additional condition (limit does not exist or does not equal zero) for the divergence test. I appreciate all the answers/comments.
 A: Everything is correct, though the notation $\lim\limits_{n\to\infty}a_n=L_{a_n}$ is nonstandard. Perhaps an improvement would be to write it as $\lim\limits_{n\to\infty}a_n=L_{a}.$ Normally just an $L$ will suffice but if you're working with multiple sequences (such as $a_n, b_n, c_n$) then $L_a$ is a good notation for the limit of the sequence $a$.
The reason why $L_{a_n}$ is not so good is because $n$ is just a free variable that has no importance whatsoever; if you write $a_n$ or $a_k$ it's the same thing. What matters is the sequence whose name is "$a$". 
Also note that your last two statements are trivially equivalent; they are contrapositives of each other. In general "If $p$ then $q$" is stating the same thing as "If not $q$ then not $p$". 
Therefore, just from knowing

$$\sum\limits_{n=0}^{\infty}a_n \: converges \rightarrow
   \lim\limits_{n\to\infty}a_n=0.$$

you can deduce

$not (\lim\limits_{n\to\infty}a_n=0 ) \rightarrow
    not (\sum\limits_{n=0}^{\infty}a_n \: converges) $

or in other words

$\lim\limits_{n\to\infty}a_n \not =0 \text{ or the limit doesn't exist} \rightarrow \sum\limits_{n=0}^{\infty}a_n \: diverges$ 

A: Everything you wrote looks good to me. 
