Power Series — Question 6

*Question 6 Determine the radius and interval of convergence of $\displaystyle\sum_{n=0}^{\infty}x^n/n!$. Explain why the factorial denominator allows every real $x$.

*Question 6 – Solution

$$\left|\frac{x^{n+1}/(n+1)!}{x^n/n!}\right|=\frac{|x|}{n+1}\longrightarrow0.$$ For every fixed real $x$, the limit is $0<1$.

The series converges absolutely for all $x\in\mathbb R$. Thus $$\boxed{R=\infty},\qquad \boxed{(-\infty,\infty)}.$$ There are no finite endpoints to test.

Each ratio contains the growing divisor $n+1$, which eventually dominates any fixed $|x|$. The series is the familiar expansion $e^x=\sum x^n/n!$.