Results 1 to 6 of 6

Math Help - decreasing sequence convergence

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    2

    decreasing sequence convergence

    Suppose that {an} is a decreasing sequence with
    lim an =0.

    Show that {an} converges if and only if {2^n a(2n)} converges.
    (as n goes to infinity)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member TheAbstractionist's Avatar
    Joined
    Apr 2009
    Posts
    328
    Thanks
    1
    Quote Originally Posted by flash89 View Post
    Suppose that {an} is a decreasing sequence with
    lim an =0.

    Show that {an} converges if and only if {2^n a(2n)} converges.
    (as n goes to infinity)
    Hi flash89.

    I think you may have stated your question incorrectly. If a_n=\frac1n then \left(a_n\right) and decreasing and converges but \left(2^na_{2n}\right)=\left(\frac{2^{n-1}}n\right) diverges.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jun 2009
    Posts
    2
    Right sorry i meant,
    Suppose that {an} is a decreasing sequence with
    lim an =0.

    Show that the sum of {an} converges if and only if {2^n a(2n)} converges.
    (as n goes to infinity)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by flash89 View Post
    Right sorry i meant,
    Suppose that {an} is a decreasing sequence with
    lim an =0.

    Show that the sum of {an} converges if and only if {2^n a(2n)} converges.
    (as n goes to infinity)
    That still can't be right, as it only places a restriction on the even terms, so the odd terms can be 1/(2k+1) and then the sum diverges whatever the even terms are doing if they are all positive (or if the sum of the even terms converges).

    CB
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member TheAbstractionist's Avatar
    Joined
    Apr 2009
    Posts
    328
    Thanks
    1
    Quote Originally Posted by CaptainBlack View Post
    That still can't be right, as it only places a restriction on the even terms, so the odd terms can be 1/(2k+1) and then the sum diverges whatever the even terms are doing if they are all positive (or if the sum of the even terms converges).
    But \left(a_n\right) is a decreasing sequence, isnít it? Then each odd term has be lie between consecutive even terms.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by TheAbstractionist View Post
    But \left(a_n\right) is a decreasing sequence, isnít it? Then each odd term has be lie between consecutive even terms.
    Opps .. missed that.

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. does the sequence necessarily become decreasing??
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: December 30th 2011, 01:28 AM
  2. prove sequence is decreasing using PMI
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 8th 2011, 11:15 PM
  3. Decreasing sequence Aleph's
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: December 25th 2009, 01:19 PM
  4. Show the sequence is decreasing
    Posted in the Calculus Forum
    Replies: 4
    Last Post: October 2nd 2008, 10:24 PM
  5. Show a sequence is decreasing
    Posted in the Calculus Forum
    Replies: 13
    Last Post: September 24th 2008, 12:59 PM

Search Tags


/mathhelpforum @mathhelpforum