Results 1 to 9 of 9

Math Help - series

  1. #1
    Member tinspire's Avatar
    Joined
    Jul 2014
    From
    cali
    Posts
    99
    Thanks
    14

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


    please help!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,489
    Thanks
    963

    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member tinspire's Avatar
    Joined
    Jul 2014
    From
    cali
    Posts
    99
    Thanks
    14

    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?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member tinspire's Avatar
    Joined
    Jul 2014
    From
    cali
    Posts
    99
    Thanks
    14

    Re: series

    is the interval [-1/6,1/6)?????
    Last edited by tinspire; July 26th 2014 at 06:17 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,489
    Thanks
    963

    Re: series

    Quote Originally Posted by tinspire View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member tinspire's Avatar
    Joined
    Jul 2014
    From
    cali
    Posts
    99
    Thanks
    14

    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?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,489
    Thanks
    963

    Re: series

    Quote Originally Posted by tinspire View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member tinspire's Avatar
    Joined
    Jul 2014
    From
    cali
    Posts
    99
    Thanks
    14

    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 $LaTeX$
    Last edited by tinspire; July 26th 2014 at 09:10 PM.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member tinspire's Avatar
    Joined
    Jul 2014
    From
    cali
    Posts
    99
    Thanks
    14

    Re: series

    in converges in the interval [-1,1)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: December 9th 2012, 12:50 PM
  2. Replies: 2
    Last Post: May 22nd 2012, 05:57 AM
  3. Replies: 3
    Last Post: September 29th 2010, 06:11 AM
  4. Replies: 0
    Last Post: January 26th 2010, 08:06 AM
  5. Replies: 1
    Last Post: May 5th 2008, 09:44 PM

Search Tags


/mathhelpforum @mathhelpforum