Question

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

Answer 1

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,
`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}`.