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&amp;\lim_{n\to\infty}\sum_{j=1}^n\\
(2)\qquad&amp;\sum_{j=1}^\infty\end{align}
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.
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.