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Math Help - Another Convergence Problem

  1. #1
    Member RedBarchetta's Avatar
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    Another Convergence Problem

    Let \sum\limits_{n = 1}^\infty  {a_n } be a convergent series of positive terms.

    What can be said about the convergence of \sum\limits_{n = 1}^\infty  {\frac{{a_1  + a_2  +  \cdots  + a_n }}<br />
{n}} ?

    My Attempt:

    <br />
\because \sum\limits_{n = 1}^\infty  {a_n }  = s_n \therefore \sum\limits_{n = 1}^\infty  {\frac{{a_1  + a_2  +  \cdots  + a_n }}<br />
{n}}  = \sum\limits_{n = 1}^\infty  {\frac{{s_n }}<br />
{n}}  = s_n \sum\limits_{n = 1}^\infty  {\frac{1}<br />
{n}} <br />


    Therefore, <br />
\sum\limits_{n = 1}^\infty  {\frac{{a_1  + a_2  +  \cdots  + a_n }}<br />
{n}} <br />
is a divergent harmonic series?
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi,
    Quote Originally Posted by RedBarchetta View Post
    \sum\limits_{n = 1}^\infty  {\frac{{s_n }}<br />
{n}}  = s_n \sum\limits_{n = 1}^\infty  {\frac{1}<br />
{n}} <br />
    You can't do this, s_n depends on n !

    Hint : for all n\geq 1, s_n\geq a_1>0 because a_k>0 for all  k\geq 1.
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by RedBarchetta View Post
    Let \sum\limits_{n = 1}^\infty {a_n } be a convergent series of positive terms.

    What can be said about the convergence of \sum\limits_{n = 1}^\infty {\frac{{a_1 + a_2 + \cdots + a_n }}<br />
{n}} ?

    My Attempt:

    <br />
\because \sum\limits_{n = 1}^\infty {a_n } = s_n \therefore \sum\limits_{n = 1}^\infty {\frac{{a_1 + a_2 + \cdots + a_n }}<br />
{n}} = \sum\limits_{n = 1}^\infty {\frac{{s_n }}<br />
{n}} = s_n \sum\limits_{n = 1}^\infty {\frac{1}<br />
{n}} <br />


    Therefore, <br />
\sum\limits_{n = 1}^\infty {\frac{{a_1 + a_2 + \cdots + a_n }}<br />
{n}} <br />
is a divergent harmonic series?
    What about this \sum_{n=1}^{\infty}\frac{a_1+a_2+\cdots+a_n}{n}\si  m\sum_{n=1}^{\infty}\frac{a_n}{n}\leqslant\sum_{n=  1}^{\infty}a_n
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  4. #4
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    \sum_{n=1}^{\infty}\frac{a_1+a_2+\cdots+a_n}{n}\si  m\sum_{n=1}^{\infty}\frac{a_n}{n}
    What do you mean by \sim ?
    Last edited by flyingsquirrel; November 15th 2008 at 01:32 PM. Reason: ...
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by flyingsquirrel View Post
    What do you mean by \sim ?
    Sorry, I misread the problem. But here is how I would present the solution

    Since this is a decreasing sequence of positive numbers let us apply Cauchy 'sCondensation test.

    \sum_{n=1}^{\infty}\sum_{k=1}^{n}a_k\frac{1}{n} converges iff \sum_{n=1}^{n}2^n\cdot\sum_{k=1}^{2n}a_k\frac{1}{2  ^n}=\sum_{n=1}^{\infty}\sum_{k=1}^{2^n}a_k converges. From there it should be pretty obvious what the answer is.
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