# How do you find the radius of convergence of sum_(n=0)^oox^n ?

Sep 2, 2014

By Ratio Test, we can find the radius of convergence: $R = 1$.

By Ratio Test, in order for ${\sum}_{n = 0}^{\infty} {a}_{n}$ to converge, we need
$\setminus {\lim}_{n \to \infty} | \frac{{a}_{n + 1}}{{a}_{n}} | < 1$.

For the posted power series, ${a}_{n} = {x}^{n}$ and ${a}_{n + 1} = {x}^{n + 1}$.
So, we have
$\setminus {\lim}_{n \to \infty} | \frac{{x}^{n + 1}}{{x}^{n}} | = {\lim}_{n \to \infty} | x | = | x | < 1 = R$

Hence, its radius of convergence is $R = 1$.