application of binomial theorem

Question

I read the following in a textbook and can't understand how the binomial theorem is being applied here:

$x_n=n^{1/n}-1$. Then $x_n\ge 0$ and by binomial theorem, $$n=(1+x_n)^n\ge \frac{n(n-1)}{2}x_n^2$$

Answer 1

The binomial theorem says that $$ (1 + x_n)^n = \sum_{k = 0}^n \binom{n}{k} x_n^k $$ which is greater than just the term when $k = 2$: $$ \binom{n}{2} x_n^2 $$ since $x_n \geq 0$. Notice now that $$ \binom{n}{2} = \dfrac{n!}{2!(n-2)!} = \frac{n(n-1)}{2}, $$ so we get $$ (1 + x_n)^n = \sum_{k = 0}^n \binom{n}{k} x_n^k \geq \binom{n}{2} x_2^2 = \frac{n(n-1)}{2} x_n^2. $$