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}#.