I'm currently stuck with the following problem:
Find all the roots of the equation $$1-\frac{x}{1}+ \frac{x(x-1)}{2!}-...+(-1)^n \frac{x(x-1)...(x-n+1)}{n!}=0$$
I can sort of see that the roots would be $1,2,3,...,n$ but cannot find a way to actually calculate them or show that they indeed are the roots. It'd be great if anyone can give me any suggestions. Hints would be more welcome than full answers. Thank you.
Call $f(x)$ your polynomial. For all $a \in \{ 1, \dots , n\}$ you have $$f(a) = \sum_{k=0}^a \binom{a}{k} (-1)^k + 0 \times \mbox{something} = \sum_{k=0}^a \binom{a}{k} (-1)^k1^{a-k} = (1-1)^a=0^a=0$$ hence, $1, \dots , n$ are roots of $f(x)$.
Since $f$ has at most $n$ roots, we found them all.