Negative value for k in binomial theorem

Question

Can $k$ in the binomial theorem be less than zero so that there is a term that essentially becomes $\dfrac{x^{n-k}}{y^{-k}}$?

Answer 1

The most general form of the binomial theorem is: $$ (1+x)^z=\sum_{k\ge0}\binom zk x^k,\tag1 $$ where $x $ and $z $ are complex numbers, $|x|\le1$, and the binomial coefficient is defined as $$ \binom zk =\frac1 {k!}\prod_{i=0}^{k-1} (z-i). $$

The point is that even in this most general case $k $ can take only non-negative integer values. This fact can be traced back to the intention of $(1) $ to represent the left hand side as power series of $x $.