Power Series — Question 5

*Question 5 Determine the radius and set of convergence of $\displaystyle\sum_{n=0}^{\infty}n!x^n$. Use the Ratio Test to show what happens for every nonzero $x$.

*Question 5 – Solution

With $a_n=n!x^n$, $$\left|\frac{a_{n+1}}{a_n}\right|=(n+1)|x|\longrightarrow\infty.$$ The Ratio Test therefore gives divergence for every fixed nonzero $x$. Indeed, the terms eventually increase in magnitude and cannot tend to zero.

At $x=0$, the series has only its $n=0$ term nonzero (using $0^0=1$ in the power-series convention), so it converges to $1$.

The radius is $\boxed{R=0}$ and the set of convergence is the single point $\boxed{\{0\}}$. This is not a nondegenerate interval.