Determine the radius of convergence of the following power series.
a) $\sum_{n=1}^{\infty}\frac{ x^{6n+2}}{(1+\frac{1}{n})^{n^2}}$
my attempts: by applying the ratio test i got $ \frac {a_n}{a_{n+1}}$ =$\frac{ x^{6n+2}}{(1+\frac{1}{n})^{n^2}}$.$\frac{(1+\frac{1}{n+1})^{(n+1)^2}}{ x^{6n+8}}$
i got $ \frac {a_n}{a_{n+1}}$ = $\frac{e}{x^6}$
now i don't know …how to find the radius of convergence of given power series…..Pliz help me
thanks in advance
Best Answer
From $\frac {a_n}{a_{n+1}} \to \frac{e}{x^6}$ we get
$\frac {a_{n+1}}{a_{n}} \to \frac{x^6}{e}$ .
The ratio test shows that the power series converges for $|x|<e^{1/6}$ and diverges for $|x|>e^{1/6}$, hence the radius of convergence is $e^{1/6}$.