Results 1 to 5 of 5

Math Help - How do I show the series of an*bn converges?

  1. #1
    LHS
    LHS is offline
    Member
    Joined
    Feb 2009
    From
    Oxford
    Posts
    84

    How do I show the series of an*bn converges?



    How do I show the series of an*bn converges?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433
    |ab|\le a^2+b^2
    Follow Math Help Forum on Facebook and Google+

  3. #3
    LHS
    LHS is offline
    Member
    Joined
    Feb 2009
    From
    Oxford
    Posts
    84
    how about

    0<=(a-b)^2
    2ab<=a^2+b^2
    2ab<=a^2+b^2
    therefore

    2*sum(ab)<=sum(a^2)+sum(b^2)

    Where the RHS converges

    As the LHS converges absolutely sum(ab) converges?

    Any ideas about the examples?


    this is from a past paper by the way, I can link you if you like, it's not a take home test
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    hint:

    Consider the series

    a_n=\left( \frac{i}{\sqrt{n}} \right)^n

    Notice that

     (a_n)^2 = \frac{(-1)^n}{n}

    b_n=\left( \frac{-i}{\sqrt{n}} \right)^n

    (b_n)^2=\frac{(-1)^n}{n}

    but ...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    LHS
    LHS is offline
    Member
    Joined
    Feb 2009
    From
    Oxford
    Posts
    84
    Ah, but an*bn=1/n.. and taking the the sum of 1/n we find by the integral test that is diverges! And by the Alternating series test (-1)^n/n converges
    Thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: October 3rd 2011, 01:12 AM
  2. Show that sin(a_n) converges
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: September 7th 2010, 11:16 AM
  3. How to show this converges to 0
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 15th 2009, 01:35 PM
  4. Show a function series converges
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 13th 2009, 08:40 AM
  5. Show a series converges for any x
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 24th 2006, 11:29 PM

Search Tags


/mathhelpforum @mathhelpforum