Results 1 to 3 of 3

Math Help - Series help!; converge or diverge

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    3

    Series help!; converge or diverge

     \sum_{n=1} ^\infty (\sqrt[3]{n})^{3n} * {(\Pi-2)}^{-4n^2}<br />

    Does this converge or diverge and how?



    \sum_{n=1} ^\infty {sinh(1/e^x)}

    I'm leaning on converge for this one because

    the limit of (1/e^x) = 0
    so,

    \sum_{n=1} ^\infty {sinh(1/e^x)}
    Converges by the Limit Comparison Test?
    Last edited by badBKO; November 23rd 2008 at 05:27 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by badBKO View Post
     \sum_{n=1} ^\infty (\sqrt[3]{n})^{3n} * {(\Pi-2)}^{-4n^2}<br />

    Does this converge or diverge and how?



    \sum_{n=1} ^\infty {sinh(1/e^x)}

    I'm leaning on converge for this one because

    the limit of (1/e^x) = 0
    so,

    \sum_{n=1} ^\infty {sinh(1/e^x)}
    Converges by the Limit Comparison Test?
    \sum_{n=1}^{\infty}\frac{n^n}{(\pi-2)^{4n^2}}

    What about the Root test?

    \begin{aligned}\lim_{n\to\infty}\sqrt[n]{\frac{n^n}{(\pi-2)^{4n^2}}}&=\lim_{n\to\infty}\frac{n}{(\pi-2)^{4n}}\\<br />
&=0<1\end{aligned}

    The last part was gotten since the exponential function increases faster than any polynomial. So this series is convergent

    \sum_{n=0}^{\infty}\sinh\left(e^{-x}\right)

    Right idea! consider

    \lim_{x\to\infty}\frac{\sinh\left(e^{-x}\right)}{e^{-x}}

    Making the sub z=e^{-x} gives

    \begin{aligned}\lim_{z\to{0}}\frac{\sinh(z)}{z}&=\  lim_{z\to{0}}\frac{\sin(iz)}{iz}\\<br />
&=1\end{aligned}

    The last part can be made with the sub iz=\varphi

    So both \sum_{n=0}^{\infty}\frac{1}{e^x} and \sum_{n=0}^{\infty}\sinh\left(\frac{1}{e^x}\right) share convergence.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2008
    Posts
    3
    Thanks alot!!!!

    The first one didn't look that easy, the second one, I thought I was on the right track.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Does this series converge or diverge?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 6th 2010, 10:15 AM
  2. Series: Converge of Diverge
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: May 6th 2010, 05:05 AM
  3. Does this series converge or diverge?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: October 30th 2009, 08:57 AM
  4. Series converge or diverge?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: June 1st 2008, 09:12 AM
  5. More on series converge or diverge?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 21st 2008, 03:28 AM

Search Tags


/mathhelpforum @mathhelpforum