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Math Help - Series convergence/divergence

  1. #1
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    Series convergence/divergence

    Hello,

    I am supposed to show why the following series are convergent or divergent, but I didn't get very far with either direct comparison, root, or ratio tests:


    1.
    \sum\limits_{n = 0}^\infty {\frac{{1}}<br />
{{{n^2} + a^2 }}}

    I would have thought that by direct comparison I could show that (1) converges, but not sure. (2) leaves me at a complete loss, though.
    2.
    \sum\limits_{n = 0}^\infty {n  \left(\frac{{{2n+1} }}<br />
{{{3n+1}}}\right)^n}

    any help appreciated.

    thank you -
    Oz.
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  2. #2
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    you should use a direct comparison test for the first one, compare it with a p series... can you see which one? When you add something to the denominator the fraction gets smaller
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  3. #3
    Super Member General's Avatar
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    Quote Originally Posted by ozfingwe View Post
    Hello,

    I am supposed to show why the following series are convergent or divergent, but I didn't get very far with either direct comparison, root, or ratio tests:


    1.
    \sum\limits_{n = 0}^\infty {\frac{{1}}<br />
{{{n^2} + a^2 }}}

    I would have thought that by direct comparison I could show that (1) converges, but not sure. (2) leaves me at a complete loss, though.
    2.
    \sum\limits_{n = 0}^\infty {n \left(\frac{{{2n+1} }}<br />
{{{3n+1}}}\right)^n}

    any help appreciated.

    thank you -
    Oz.
    1.
    n^2+a^2 \geq n^2 for every a \in R
     \frac{1}{n^2+a^2} \leq \frac{1}{n^2}
    Now, You are supposed to complete it.

    2.
    After applying tha Root Test, you will face the following limit:
    \frac{2}{3}\lim_{n\to\infty} n^{\frac{1}{n}} .
    Evaluate this limit, and hance, determine the convergence of the series by using the concept of the Root Test.


    **Edit** :
    Ohhh sorry I did not read the whole question, Try to use the integral test to the first one.
    What is the tests which you are fimilar with ?
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  4. #4
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    Quote Originally Posted by General View Post
    1.
    n^2+a^2 \leq n^2 for every a \in R
     \frac{1}{n^2+a^2} \geq \frac{1}{n^2}
    Now, You are supposed to complete it.

    2.
    After applying tha Root Test, you will face the following limit:
    \frac{2}{2}\lim_{n\to\infinity} n^{\frac{1}{n}} .
    Evaluate this limit, and hance, determine the convergence of the series by using the concept of the Root Test.

    you swapped your \leq and \geq in the first

    proof: let a=1, n^2+1\not\leq n^2
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  5. #5
    Super Member General's Avatar
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    Quote Originally Posted by artvandalay11 View Post
    you swapped your \leq and \geq in the first

    proof: let a=1, n^2+1\not\leq n^2
    Thanks.
    Acuatlly I mixed the \geq and \leq =)
    Its Latex Typo =D

    It has been edited.
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