Results 1 to 6 of 6

Thread: disprove a convergence question?

  1. #1
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    disprove a convergence question?

    i know that An->1

    i need to prove that (An)^n ->1

    but when i construct limit
    lim (An)^n
    n->+infinity

    i get 1^(+infinity) which says that there is no limit
    what do i do in this case in order to disprove that (An)^n->1

    ??
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,743
    Thanks
    2814
    Awards
    1
    It is true that $\displaystyle \left( {A_n = \sqrt[n]{n}} \right) \to 1$.
    But what about $\displaystyle \left( {\sqrt[n]{n}} \right)^n \to ?$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    where did you find the power of 1/n
    when i do the limit
    the base goes to 1 and the power goes to + infinity

    that is not solvable

    what to do??

    they question says prove/disprove

    how to disprove
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by transgalactic View Post
    i know that An->1

    i need to prove that (An)^n ->1

    but when i construct limit
    lim (An)^n
    n->+infinity

    i get 1^(+infinity) which says that there is no limit
    what do i do in this case in order to disprove that (An)^n->1

    ??
    Why not try this method? Your book ought to show that the Root and Ratio test always yield the same result. So

    $\displaystyle \limsup \sqrt[n]{A_n}=\limsup\frac{A_{n+1}}{A_n}$

    And you have already asked this question already.

    Also let us try a three case scenario again

    Case #1: $\displaystyle A_n\geqslant A_{n+1}\cdots$

    It is clear then from the fact that $\displaystyle A_n\to1$ that there exists some $\displaystyle N$ such that [tex]N\leqslant{n}[/math[ implies

    $\displaystyle A_n\geqslant{1}$

    From there it is clear then that

    $\displaystyle 1\leqslant{A_n}\leqslant{A_n^n}$

    Or $\displaystyle 1\leqslant\sqrt[n]{A_n}\leqslant{A_n}$


    Case #2: $\displaystyle A_n\leqslant{A_{n+1}}\cdots$

    From here it is clear that there exists a $\displaystyle N$ such that $\displaystyle N\leqslant{n}$ implies $\displaystyle 0\leqslant{A_n}\leqslant{1}$

    And it should be clear then that $\displaystyle A_n^n\leqslant{A_n}\leqslant{1}$

    Or

    $\displaystyle A_n\leqslant\sqrt[n]{A_n}\leqslant{1}$


    Case #3: This is when $\displaystyle A_n=A_{n+1}\cdots$ where the conclusion readily follows.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    how did you came to the conclusion that my limit equal this?

    $\displaystyle

    \limsup \sqrt[n]{A_n}=\limsup\frac{A_{n+1}}{A_n}$
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by transgalactic View Post
    how did you came to the conclusion that my limit equal this?

    $\displaystyle

    \limsup \sqrt[n]{A_n}=\limsup\frac{A_{n+1}}{A_n}$
    Im sorry when I saw Plato's post I mistakenly believed you were asking to prove that $\displaystyle A_n\to1\implies\sqrt[n]{A_n}\to{1}$, forgive me.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. disprove convergence
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Dec 12th 2011, 02:26 PM
  2. Prove/disprove convergence of two series.
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Oct 18th 2011, 11:26 PM
  3. Prove or disprove question
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Nov 7th 2010, 09:19 PM
  4. Prove or disprove convergence of an infinite series
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Oct 17th 2010, 11:08 AM
  5. prove or disprove question
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: Sep 17th 2010, 10:21 AM

Search Tags


/mathhelpforum @mathhelpforum