What does the sum notation $\sum\limits_{j=n}^\infty a_j$ mean? What does the sum notation $\sum\limits_{j=n}^\infty a_j$, $n\in\mathbb{N}$, mean?


Is this the sequence of the sums, i.e:

$$ \sum_{j=n}^\infty a_j=\left( \sum_{j=1}^\infty a_j,\;
  \sum_{j=2}^\infty a_j,\;
  \sum_{j=3}^\infty a_j,\;
  \sum_{j=4}^\infty a_j,\;
  \sum_{j=5}^\infty a_j,\;
   \ldots\right)\ ?
$$
It means the limit of the partial sums when $k\rightarrow \infty$ and $n\in\mathbb N$ is fixed, i.e $$\lim\limits_{k\rightarrow\infty}\sum\limits_{j=n}^{k}{a_j}$$
If it was $$\left\{\sum\limits_{j=n}^{\infty}{a_j}\right\}_{n=1}^{\infty}$$ then it is the sequence that you wrote.
Basically the notation
$$\sum\limits_{j=n}^\infty a_j$$ is the infinite sum of the values of $a_j$ when $j$ changes so:
$$
\sum\limits_{j=n}^\infty a_j=a_n+a_{n+1}++a_{n+2}+a_{n+3}+a_{n+4}+ \cdots
$$ up until $n$ reaches infinity.