Math Help - series

1. series

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 $\sum_{n=1}^{\infty} \frac{8^nx^n}{(n+5)^2}$

2. Re: series

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.

3. Re: series

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?

4. Re: series

is the interval [-1/6,1/6)?????

5. Re: series

Originally Posted by tinspire
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?
I think you're wrong, unless you meant to write $(6x)^n$ instead of $6x^n$

what makes you think a convergence interval must be closed? In almost all cases it won't be.

6. Re: series

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?

7. Re: series

Originally Posted by tinspire
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?
without the correct use of parentheses or at least superscripting I just can't interpret what you've written above.

8. Re: series

ok nevermind your right. i made a mistake. it is $|x|$ not $6|x|$. sorry about the last post. i was just feeling to lazy to do it in $LaTeX$

9. Re: series

in converges in the interval [-1,1)