# How do you find the radius of convergence for a power series?

Sep 23, 2014

To demonstrate, let us find the radius of convergence of

${\sum}_{n = 0}^{\infty} \frac{{\left(x - 4\right)}^{2 n}}{{3}^{n}}$.

By Ratio Test,

lim_{n to infty}|{{(x-4)^{2n+2}}/{3^{n+1}}}/{{(x-4)^{2n}}/{3^n}}| =lim_{n to infty}|{(x-4)^2}/{3}|={|x-4|^2}/3<1

by multiplying by 3,

$R i g h t a r r o w | x - 4 {|}^{2} < 3$

by taking the square-root,

$R i g h t a r r o w | x - 4 | < \sqrt{3} = R$

Hence, the radius of convergence is $R = \sqrt{3}$.