Proof: convergence about power series

Question

$\sum a_n x^n$ power series If the coefficients $a_n$ equal the answers of a family of counting problems indexed by n we call the power series a generating function for the counting problem. $1 + x + 2x^2 + 6x^3 + 24x^4 +$ equals the generating function for the problem of counting ways to order an n element set

How to prove that this never converges for any x except 0?

Answer 1

$$1+x+2x^2+6x^3+24x^4+120x^5+\cdots=\\ \sum_{n=0}^{\infty}n!\times x^{n}\\$$suppose $|x|<1$ now take $$x=\frac{1}{k} , \space |k|>1$$ $$\lim_{n\to \infty}a_n=\\\lim_{n\to \infty}n!x^n=\\\lim_{n\to \infty} \frac{n!}{k^n} \to \infty$$