What is the difference between these two sets of symbols? I am considering two sets of symbols. Are these two sets different or the same? Why?
\begin{align}(1)\qquad&\lim_{n\to\infty}\sum_{j=1}^n\\
(2)\qquad&\sum_{j=1}^\infty\end{align}
 A: 
The series (2) in the form 
  $\sum_{j=1}^{\infty}a_n$
  admits two different interpretations.
  
  
*
  
*It can be seen as sequence  of partial sums 
  \begin{align*}
\sum_{j=1}^{\infty}a_j:=\left(a_1,a_1+a_2,a_1+a_2+a_3,\ldots\right)=\left(\sum_{j=1}^na_j\right)_{n\geq 1}
\end{align*}
  
*and it can be seen as limit of the sequence of partial sums
  \begin{align*}
\sum_{j=1}^{\infty}a_j:=\lim_{n\to\infty}\sum_{j=1}^na_j
\end{align*}
In the second case the symbols (1) and (2) have the same meaning whereby (2) is defined by (1).

Note: You can find this line of argumentation e.g. in K. Knopps Theory and application of infinite series.
A: Lets write these out carefully for some sequence of numbers $a_n$.
The first thing you wrote is $$\lim_{n\to\infty}\sum_{j=1}^na_n.$$ This is already pretty well defined (if you are comfortable with the definition of a limit of a sequence, since that is exactly what this is).
The second thing is $$\sum_{j=1}^\infty a_n.$$ This does not appear to be a limit (at least not explicitly), however we cannot actually keep adding infinitely many terms to each other, so there must be some limit involved. This hidden limit is precisely the first thing you wrote, and as mentioned in the comments the first expression is the definition of this second expression.
A: I guess in some formal setting the sum $$\sum_{j=1}^{\infty}()$$ 
is defined to be $$\lim_{n \to \infty} \sum_{j=1}^{n}()$$, how else can we take infinite sums or prove results? With the limit. Indeed if you "believe" in infinity or not the results can be positively guaranteed.
