TITLE: What does the sum notation $\lim_{n\to\infty} \sum_{j=n}^\infty a_j$ mean?
QUESTION [5 upvotes]: What does the sum notation$$\lim_{n\to\infty}  \sum_{j=n}^\infty a_j$$
mean?
Is the $n$ fixed or not?

REPLY [6 votes]: It is the limit of the sequence
$$\sum_{j=1}^\infty a_j,\;
  \sum_{j=2}^\infty a_j,\;
  \sum_{j=3}^\infty a_j,\;
  \sum_{j=4}^\infty a_j,\;
  \sum_{j=5}^\infty a_j,\;
   \cdots
$$
My first guess would be, if   $\sum_{j=1}^\infty a_j$ exists, then your answer has to be $0$.
For example
$$\lim_{n\to\infty} \sum_{j=n}^\infty \frac{1}{2^j} =
  \lim_{n\to\infty} \frac{2}{2^n} =
  0
$$
THEOREM. If
$\sum_{j=1}^\infty a_j$ exists, then
$\lim_{n\to\infty} \sum_{j=n}^\infty a_j = 0$
PROOF. For convenience, let $\sum_{j=1}^\infty a_j = L$. Then, since the limit exists,
$L = \sum_{j=1}^{n-1} a_j + \sum_{j=n}^\infty a_j$
Now let $n \to \infty$. Then $L = L + \lim_{n\to\infty}\sum_{j=n}^\infty a_j$
So $\lim_{n\to\infty}\sum_{j=n}^\infty a_j = 0$.