$ \ sum \ limits _ {I =1} na _ I = a _1+a _ 2+\ cdots+a _ n USD.
$ \ prod \ limits _ {I =1} na _ I = a _1\ times a _ 2 \ times \ cdots \ times a_n dollars.
In most cases, summation symbols and multiplication symbols are introduced according to intuitive ideas. However, in the formal sense, their definitions are given by the concept of limit.
Let's take the ampersand as an example to illustrate. Suppose there is a series $\{a_n\}$, and define a new series $\{s_n\}$ to satisfy $ s _ n = a _1+a _ 2+\ cdots+a _ n $,where $n$ is a positive integer. If the sequence $\{s_n\}$ has the limit $ \ displaystyle \ lim _ {n \ to \ infty} s _ n = s $,then we call the limit of the sequence $\{s_n\}$ the sum of the sequence $\{a_n\}$.
Similarly, the sign of multiplication can also be given by the definition of limit. However, due to the need to use more advanced concepts such as continuity, I will not explain them in detail here.
In a word, the derivation and definition of summation symbol and multiplication symbol need further mathematical knowledge, including limit, sequence and so on, which can only be mastered through deeper mathematical study.