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Math Help - infinite series question

  1. #1
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    infinite series question

    First of all, I want to say sorry for the language, I thing I speak pretty well English, but mathematics English is an all different game for me...
    I'm from Israel BTW.

    Ok, now for the questions:
    1. I have 2 infinite series, a^2 from 1 to infinite and b^2 from 1 to infinite, both of them converges, I need to prove the infinite series (a * b) from 1 to infinite Absolute convergence

    2.
    I have the infinite series a^2 from 1 to infinite converges, I need to prove the infinite series (a/n) from 1 to infinite also converges.

    I hope you got it, this are the last 2 question I got left in this homework.

    10x in advance.
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  2. #2
    Super Member PaulRS's Avatar
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    1. By Cauchy-Schwarz we have: <br />
\left( {\sum\limits_{k = 1}^n {a_k ^2 } } \right) \cdot \left( {\sum\limits_{k = 1}^n {b_k ^2 } } \right) \geqslant \left( {\sum\limits_{k = 1}^n {\left| {a_k  \cdot b_k } \right|} } \right)^2 <br /> <br />
what do you conclude?

    2. Again: <br />
\left( {\sum\limits_{k = 1}^n {a_k ^2 } } \right) \cdot \left( {\sum\limits_{k = 1}^n {\tfrac{1}<br />
{{k^2 }}} } \right) \geqslant \left( {\sum\limits_{k = 1}^n {\left| {\tfrac{{a_k }}<br />
{k}} \right|} } \right)^2 <br />

    And: <br />
\sum\limits_{k = 1}^{ + \infty } {\tfrac{1}<br />
{{k^2 }}} <br />
converges
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  3. #3
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    Quote Originally Posted by asi123 View Post
    1. I have 2 infinite series, a^2 from 1 to infinite and b^2 from 1 to infinite, both of them converges, I need to prove the infinite series (a * b) from 1 to infinite Absolute convergence

    2. I have the infinite series a^2 from 1 to infinite converges, I need to prove the infinite series (a/n) from 1 to infinite also converges.
    1)If \sum_{n=1}^{\infty}a_n^2 converges and \sum_{n=1}^{\infty}b_n^2 converges then \sum_{n=1}^{\infty} a_n^2b_n^2 converges but then \sum_{n=1}^{\infty}\sqrt{a_n^2b^2} converges. Now note \sqrt{a_n^2b_n^2} = |a_nb_n|.

    2) \left| \frac{a_n}{n} \right| \leq a_n^2 + \frac{1}{n^2}.

    EDIT: Paul was fast.
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