I was going over some power series problems when I noticed this one :- $ \sum \frac {3^n}{\sqrt{n}}. x^{2n+1}$ find the radius of convergence of the power Series.
Immediately i noticed that I needed to be careful while applying root test as this is a kind of Sub Sequential Power Series ! So instead of finding $n$th root we have to find the $2n+1$th root of the corresponding coefficient and then take the LimSup of it !
But i can't finish the computation of the limit ! Can you be kind enough to help me through ?
Best Answer
Use the Ratio Test instead: $$\lim_{n \longrightarrow+\infty}|\frac{a_{n+1}}{a_n}|=\lim_{n \longrightarrow+\infty}|\frac{\frac{3^{n+1}}{\sqrt{n+1}}x^{2(n+1)+1}}{\frac{3^n}{\sqrt{n}}x^{2n+1}}|=\lim_{n \longrightarrow+\infty}3x^2\sqrt\frac{n}{n+1}=3x^2<1 \Longrightarrow |x| < \frac{1}{\sqrt{3}}$$ So it's radius is $\frac{1}{\sqrt{3}}$.