find the interval of convergence of the series $\sum_{n=1}^{\infty} \frac{6x^n}{\sqrt[5]{n}}$
find the radius of convergence of the series $\sum_{n=1}^{\infty} \frac{8^nx^n}{(n+5)^2}$
please help!
for the first, using the ratio test
$L=\displaystyle{\lim_{n\to\infty}}\left |\dfrac{\dfrac{6x^{n+1}}{\sqrt[5]{n+1}}}{\dfrac{6x^n}{\sqrt[5]{n}}}\right |$
$L=\displaystyle{\lim_{n\to\infty}}\left |x \dfrac{\sqrt[5]{n}}{\sqrt[5]{n+1}}\right|$
$L=|x|$
Note also that $L < |x|, \forall n \in \mathbb{N}$
For convergence we must have $L < 1$
so the radius of convergence is
$|x| < 1$
for the second also use the ratio test, the problem is very similar to the first. I leave you to finish the details.
for the first one the answer is actually 6x. so i understand that the series converges in the interval (-1/6,1/6) but how do i find the endpoints of the interval? or is that it? also, will i always have to find the radius in order to find the interval?
it's 6x^n+1 which is 6x^n*6x^1. you can cancel out the 6x^n but your still left with another 6x. right?
well by doing this you find that it converges in this interval but dont you also have to find the end points? how to test whether it should be an open or closed interval?