Results 1 to 5 of 5

Math Help - infinite series

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    18

    infinite series

    Imagine you are stacking an infinite number of spheres of decreasing radii on top of each other. The radii of the spheres are 1m, 1/2^(1/2)m, 1/3^(1/2)m, etc. . . The spheres are made of a material that weighs 1 newton per cubic meter. . .

    1.) How high is the infinite stack of spheres

    2.) What is the total surface area of all the spheres in the stack

    3.) Show that the weight of the stack is finite.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Mentia's Avatar
    Joined
    Dec 2008
    From
    Bellingham, WA
    Posts
    134
    1)

    The following series:

    2 \sum_{1 }^{ \infty  }  \left (  { \frac{ 1}{n }  }   \right ) ^{1/2}

    is divergent. The stack would be infinitely high.

    2)

    Surface Area = 4*pi*r^2

    -> Total surface area = 4 \pi \sum_{1 }^{ \infty  }   \left (  \left (  \frac{1 }{n }   \right ) ^{1/2}  \right ) ^{2} = 4 \pi \sum_{1 }^{ \infty  }  \frac{1 }{ n}

    This is the famous harmonic series. Also divergent. So surface area is infinite.

    3) You can probably see where this one will lead. Since volume is proportional to r^3, you will have an overharmonic series which will converge:

     \frac{4 }{3 }  \pi \sum_{1 }^{ \infty  }   \left (  \left (  \frac{1 }{n }   \right ) ^{1/2}  \right ) ^{3} =  \frac{ 4}{ 3}  \pi \sum_{1 }^{ \infty  } ( \frac{1 }{ n} )^{ \frac{ 3}{2 } } =  \frac{ 4}{ 3}  \pi  \zeta (3/2)  \approx 10.94  cubic meters

    -> Weight = 10.94 newtons


    Thus, infinite height, infinite surface area, finite weight.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2008
    Posts
    18
    thank you. I had a feeling that was going to be how it worked out, but i wasn't positive on how to actually show it. Thanks

    sorry one more question. I understand the p-series test has (1/n)^(3/2) converging, but i dont understand to what value it converges or how you came up with 10.94 meters
    Last edited by mr fantastic; March 8th 2009 at 04:01 AM. Reason: Merged posts
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member Mentia's Avatar
    Joined
    Dec 2008
    From
    Bellingham, WA
    Posts
    134
    Okay so check out the Riemann Zeta function as given by wikipedia:



    So in your case s = 3/2. Wikipedia gives Zeta(3/2) = 2.612

    So (4/3)*Pi*2.612 = approx 10.94
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Oct 2008
    Posts
    18
    o ok thank you
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Fourier series to calculate an infinite series
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 4th 2010, 02:49 PM
  2. Replies: 2
    Last Post: May 26th 2010, 10:17 AM
  3. Replies: 7
    Last Post: October 12th 2009, 11:10 AM
  4. Replies: 2
    Last Post: September 16th 2009, 08:56 AM
  5. Replies: 1
    Last Post: May 5th 2008, 10:44 PM

Search Tags


/mathhelpforum @mathhelpforum