Power Series — Question 6

Question 6 – Solution

Step 1: Apply the Ratio Test for a fixed $x$.

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

Step 2: State the convergence set.

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.

Step 3: Interpret the factorial.

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