Prove that the radius of convergence $R$ of $exp: \mathbb{R} \to \mathbb{R}$ is $R= \infty$.
Solution given in past exam paper mark scheme: Let $a_n= \frac{1}{n!}$, then $\frac{|a_{n+1}|}{|a_n|}=\frac{\frac{1}{(n+1)!}}{\frac{1}{n!}}=\frac{n!}{nn!}=\frac{1}{n} \to 0$ as $n \to \infty$ Thus the radius of convergence is $R= \infty$.
Am I right in saying that some of the calculation is wrong there? It should be: $\frac{|a_{n+1}|}{|a_n|}=\frac{\frac{1}{(n+1)!}}{\frac{1}{n!}}=\frac{n!}{(n+1)!}=\frac{1}{n+1} \to 0$ as $n \to \infty$.
Can I also ask how this shows that the radius of convergence is $\infty$?