# What is the radius of convergence of the series sum_(n=0)^oo(x-4)^(2n)/3^n?

Sep 3, 2014

By Ratio Test, the radius of convergence is $\sqrt{3}$.

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 $R = \sqrt{3}$.