Problem - 4317
(Generalized binomial expansion) If $a$, $b$, and $r$ are some real or complex numbers, then $$(a+b)^r = \sum_{k=0}^{\infty}\binom{r}{k}a^{r-k}b^k$$
Here, the following definition still holds when $r$ is a real or complex number: $$\binom{r}{k}=\frac{r(r-1)\cdots(r-k+1)}{1\cdot 2\cdots k}$$
Let's consider function $f(b) = (a+b)^r$ where $a$ and $r$ are constants. Taking $k^{th}$ derivative on both sides gives $$\frac{d^k}{db^k}f(b)=r(r-1)\cdots(r-k+1)(a+b)^{r-k}$$
This means $$\frac{d^k}{db^k}f(0)=r(r-1)\cdots(r-k+1)a^{r-k}$$
Applying Taylor expansion centered at $0$ gives $$(a+b)^r = \sum_{k=0}^{\infty}\frac{r(r-1)\cdots(r-k+1)a^{r-k}}{k!}b^k=\sum_{k=0}^{\infty}\binom{r}{k}a^{r-k}b^k$$