How can I define $e^x$ as the value of infinite series? I understand the definition of $e^x$ by limit. But I do not know how to come up with:
$$e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$$
without using Taylor series. more explicitly without using calculus. how do we know if a function can be expressed as series or not ?

If you take the fact that $(e^x)'=e^x$ then by taylor series, we can expand around $0$, taking into account that each $n$-th derivative at $0$ is $1$.
Then $e^x=\sum_{n=0}^\infty\frac{f^{(n)}(0)}{n!}x^n=\sum_{n=0}^\infty\frac{1}{n!}x^n=\sum_{n=0}^\infty\frac{x^n}{n!}$
Lets try this without Taylor series. A function that can be expressed by a real power series is called real analytic. All that is needed is that all derivatives are greater than or equal to $0$. Clearly this holds for $e^x$.
So let us write $e^x=\sum_{n=0}^\infty a_nx^n$ taking derivatives we get:
$\sum_{n=1}^\infty na_nx^{n-1}=\sum_{n=0}^\infty a_nx^n$
Comparing coefficients of each power of $x$ we get:
$a_{n+1}=\frac{1}{n+1}a_n$ and we know $a_0=1$ (because $e^0=1$) this gives as $a_n=\frac{1}{n!}$
Thus $e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$
