Different summation notation? In school, I learned that 
$$\sum_{k=1}^{n}f(k)=f(1)+f(2)+f(3)+ ... +f(n)$$
But in some physics or mathematics book, I saw this kind of representation such like:
$$\lim_{\Delta x\to 0} \sum_{j} f(x_j) \Delta x$$
What is the difference between these kinds of summation notation?

The    notation 
\begin{align*}
\lim_{\Delta x\to 0} \sum_{j} f(x_j) \Delta x\tag{1}
\end{align*}
is typically used for Riemann sums when introducing integration and adresses more concepts than only finite summation $$\sum_{k=1}^{n}f(k)=f(1)+f(2)+f(3)+ ... +f(n)$$

  
*
  
*Finite sum: One part of (1) is a finite sum given in a somewhat sloppy notation (but quite usual if the context is known):
  \begin{align*}
\sum_{j} f(x_j) \Delta x\tag{2}
\end{align*}
  Here we consider $x$ to be a shorthand of an $(n+1)$-tuple $x=(x_0,x_1,\ldots,x_n)$ with the property
  \begin{align*}
x_0<x_1<\cdots<x_n
\end{align*}
  The symbol $\Delta x$ (aka difference) means the difference of two consecutive elements $x_j-x_{j-1}$ of $x$ with $1\leq j\leq n$. Another more precise notation for (2)    is
  \begin{align*}
\sum_{j} f(x_j) \Delta x&=\sum_{j=1}^n f(x_j) \Delta x_j\\
&=\color{blue}{\sum_{j=1}^n f(x_j) (x_j-x_{j-1})}\tag{3}
\end{align*}
The finite sum (3)  corresponds  to   the finite sum in your question.

But there is another twist, a specific limit of this sum.

  
*
  
*Limit of sum: The notion $$\Delta x \rightarrow 0$$ addresses a specific kind of limit built from a sequence of $(n+1)$-tuples
  $$x^{(n)}=(x_0^{(n)},x_1^{(n)},\ldots,x^{(n)}_n)$$  with $n\geq 1$ requiring that the maximum absolute difference of $\Delta x$ approaches zero when taking the limit.
\begin{align*}
\Delta x\to 0\ &\ \widehat{=}\lim_{n\rightarrow \infty}\Delta x^{(n)}=0\\
&\ \widehat{=}\lim_{n\rightarrow \infty}\left(\max_{1\leq j\leq n}\left(x_j^{(n)}-x_{j-1}^{(n)}\right)\right)=0
\end{align*}

This way we require a nice, convergent behaviour of the Riemann sums necessary for the existence of the Riemann integral which is based upon this principle.
