Ratio Test — Question 6

Question 6 – Solution

Step 1: Treat $x$ as fixed.

For $a_n=x^n/n!$, $$\left|\frac{a_{n+1}}{a_n}\right|=\frac{|x|}{n+1}.$$ For $x=0$ the series plainly converges; the same final limit covers every fixed nonzero $x$.

Step 2: Take the limit.

$$L=\lim_{n\to\infty}\frac{|x|}{n+1}=0<1\qquad\text{for every fixed }x\in\mathbb R.$$

Conclusion.

The series converges absolutely for all real $x$, so its interval of convergence is $(-\infty,\infty)$. By the exponential-series identity, $$\boxed{\displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!}=e^x}.$$