Can binomial expansion of: $$(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2 + \cdots$$ be used to prove that numbers such as $\sqrt{2}$ are irrational. Also, what is the easiest way to write this expansion using sigma notation?
I had previously written it as: $$1+\sum_{n=1}^{\infty}\frac{\prod_{m=0}^{n-1}(p-m)}{n!}x^n$$ where p is the power
The binomial expansion formula is $$ (1+x)^n = \sum _0^{\infty} \binom {n}{k} x^k $$
$$(1+x)^{1/2} = \sum _0^{\infty} \binom {1/2}{k} x^k $$
The series converges for $|x|<1$ but for $x=1$ the series does not converge to $\sqrt 2$