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

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Answer 1 · 2014-09-23T00:11:19

To demonstrate, let us find the radius of convergence of

$sum_{n=0}^infty{(x-4)^{2n}}/{3^n}$ .

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,

$Rightarrow |x-4|^2<3$

by taking the square-root,

$Rightarrow |x-4|< sqrt{3}=R$

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