Results 1 to 2 of 2

Thread: analysis - 2

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    13

    analysis - 2

    How does the comparison test incorporate the cauchy criterion for convergence?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by pila0688 View Post
    How does the comparison test incorporate the cauchy criterion for convergence?
    Let $\displaystyle T_n=\sum_{n=1}^{N} c_n$

    So the comparison test says that if $\displaystyle |a_n|\leqslant c_n~{\color{red}\star}$ for all $\displaystyle N\leqslant n$ with $\displaystyle N\in\mathbb{N}$ and$\displaystyle \sum c_n$ converges, then so does $\displaystyle \sum a_n$

    Proof: $\displaystyle \sum c_n$ converges implies that $\displaystyle T_n$ converges. Now since every convergent sequence is Cauchy $\displaystyle \color{blue}\star$ it also follows that there exists some $\displaystyle K$ that if $\displaystyle K\leqslant N\leqslant M$ then $\displaystyle \left|T_M-T_N\right|<\varepsilon$. But because of how we defined $\displaystyle T_N$ we see this is equivalent to $\displaystyle \sum_{n=N}^{M}c_n<\varepsilon$. But by $\displaystyle \color{red}\star$

    $\displaystyle \sum_{n=N}^{M}|a_n|\leqslant\sum_{n=N}^{M}c_n<\var epsilon$

    And

    $\displaystyle \left|\sum_{n=N}^{M}a_n\right|\leqslant\sum_{n=N}^ {M}|a_n|\leqslant\sum_{n=N}^{M}c_n<\varepsilon$.

    From there we conclude that $\displaystyle \sum a_n$ converges

    It is clear that we used the Cauchy convergence criterion at $\displaystyle \color{blue}\star$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with Analysis please :)
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Oct 16th 2009, 04:38 AM
  2. analysis help
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Oct 8th 2009, 03:11 PM
  3. analysis help
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Jan 19th 2009, 02:38 PM
  4. analysis
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Dec 15th 2008, 03:07 PM
  5. Analysis Help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Sep 8th 2008, 05:59 PM

Search Tags


/mathhelpforum @mathhelpforum