Question

How do you find the radius of convergence of `sum_(n=0)^oox^n` ?

Answer 1

By Ratio Test, we can find the radius of convergence: `R=1`.

By Ratio Test, in order for `sum_{n=0}^{infty}a_n` to converge, we need
`\lim_[n to infty}|{a_{n+1}}/{a_n}|<1`.

For the posted power series, `a_n=x^n` and `a_{n+1}=x^{n+1}`.
So, we have
`\lim_[n to infty}|{x^{n+1}}/{x^n}|=lim_{n to infty}|x|=|x|<1=R`

Hence, its radius of convergence is `R=1`.