similarity of the expansions of $e^x$ and $(1+x)^n$

Question

Is there a reason behind why the expansion of $(1+x)^n$ looks very similar to the taylor series expansion of $e^x$?

Answer 1

The fact that$$(\forall x\in\mathbb{C}):e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n$$leads naturally to similarities.

Answer 2

You can think of it in terms of the exponential generating funtions of a given sequence: $a_0,a_1,a_2\ldots$ defined as: $$a(x):=\sum_{n=0}^{\infty}a_{n}\frac{x^{n}}{n!}.$$ When $f_{n}=1$, $\forall\, n\geq 0$ you have: $$f(x)=\sum_{n=0}^{\infty}f_{n}\frac{x^{n}}{n!}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}=e^{x}$$ When $\alpha$ is a natural number and $g_n$ is the sequence defined as: $$g_{n}=\begin{cases} \alpha(\alpha-1)\cdots(\alpha-n+1) & \alpha\leq n\\ \qquad 0 & \text{otherwise} \end{cases},\quad \forall\, n\geq 0$$ you have: $$g(x)=\sum_{n=0}^{\infty}g_{n}\frac{x^{n}}{n!}=\sum_{n=0}^{\infty}\alpha(\alpha-1)\cdots(\alpha-n+1)\frac{x^{n}}{n!}=(1+x)^{\alpha}$$