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

Question

6797 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-09-02T18:30:52

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 $R=1$ .

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