Results 1 to 3 of 3

Math Help - How to prove that a limit exists?

  1. #1
    Member Pranas's Avatar
    Joined
    Oct 2010
    From
    Europe. Lithuania.
    Posts
    81

    How to prove that a limit exists?

    Hello.

    I have to prove that an actual limit exists while n is approaching infinity.
    I do NOT need to get to the exact limit itself.

    1. It is easy to understand (and prove), that sequence never decreases. That's already half of the job.
    2. I need to make some manipulations that would lead to any specific number being greater than my sequence (irrespective of natural n value).



    I get no further than this, a remaining product is a real headache.
    Thank you for ideas and sorry for possibly broken English
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by Pranas View Post
    Hello.

    I have to prove that an actual limit exists while n is approaching infinity.
    I do NOT need to get to the exact limit itself.

    1. It is easy to understand (and prove), that sequence never decreases. That's already half of the job.
    2. I need to make some manipulations that would lead to any specific number being greater than my sequence (irrespective of natural n value).



    I get no further than this, a remaining product is a real headache.
    Thank you for ideas and sorry for possibly broken English
    \lim\limits_{n\to\infty}\prod\limits^n_{k=1}\frac{  2^k+1}{2k}=\prod\limits^\infty_{k=1}\left(1+\frac{  1}{2^k}\right) . Applying logarithm you get that the infinite product converges

    iff the infinite series \sum\limits^\infty_{k=1}\log\left(1+\frac{1}{2^k}\  right) converges (the logarithm's base never minds, of course).

    Lemma: if \{a_n\} is a positive sequence, the series \sum\limits^\infty_{k=1}\log\left(1+a_k\right) converges iff the series \sum\limits^\infty_{k=1}a_k converges.

    Proof sketch: assuming a_k\xrightarrow [k\to\infty]{}0 , we get what we want from the limit test for positive series, since \frac{\log(1+a_k)}{a_k}\xrightarrow [k\to\infty]{}1\,\,\,\,\square

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Pranas's Avatar
    Joined
    Oct 2010
    From
    Europe. Lithuania.
    Posts
    81
    Thank you very much, I was not aware of the lemma.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to prove that this limit does exists...
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 26th 2011, 08:51 AM
  2. How to prove that a limit exists at 0.
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 27th 2011, 03:22 PM
  3. Prove the limit of this sequence exists
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 14th 2010, 11:17 PM
  4. Replies: 3
    Last Post: May 14th 2009, 01:31 PM
  5. Replies: 1
    Last Post: April 11th 2009, 12:35 PM

/mathhelpforum @mathhelpforum