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Thread: Determine whether the series converges or diverges.

  1. #1
    Junior Member moonman's Avatar
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    Determine whether the series converges or diverges.

    $\displaystyle \sum_{n=1}^{\infty}\frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)} $

    Will use comparison test:

    $\displaystyle \sum a_{n} = \sum_{n=1}^{\infty}\frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)}$

    $\displaystyle \sum b_{n} = \frac{n^2}{30n^3} $

    Is this correct series for $\displaystyle \sum b_{n} $ ?
    Last edited by moonman; Nov 26th 2012 at 12:28 PM.
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  2. #2
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    Re: Determine whether the series converges or diverges.

    Can't see your series.
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  3. #3
    Junior Member moonman's Avatar
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    Re: Determine whether the series converges or diverges.

    Thanks, I had to change the latex
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  4. #4
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    Re: Determine whether the series converges or diverges.

    Quote Originally Posted by moonman View Post
    $\displaystyle \sum_{n=1}^{\infty}\frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)} $

    Will use comparison test:

    $\displaystyle \sum a_{n} = \sum_{n=1}^{\infty}\frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)}$

    $\displaystyle \sum b_{n} = \frac{n^2}{30n^3} $

    Is this correct series for $\displaystyle \sum b_{n} $ ?

    Just note that $\displaystyle \frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)}\le \frac{4n^2}{(3n+2)^2(5n+1)}$
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  5. #5
    Junior Member moonman's Avatar
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    Re: Determine whether the series converges or diverges.

    Quote Originally Posted by Plato View Post
    Just note that $\displaystyle \frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)}\le \frac{4n^2}{(3n+2)^2(5n+1)}$
    How did you get the 4 in the numerator?
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  6. #6
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    Re: Determine whether the series converges or diverges.

    Quote Originally Posted by moonman View Post
    How did you get the 4 in the numerator?

    $\displaystyle \sqrt{n^4 + 5n + 10}\le\sqrt{n^4 + 5n^4 + 10n^4}=4n^2$
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  7. #7
    Junior Member moonman's Avatar
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    Re: Determine whether the series converges or diverges.

    Quote Originally Posted by Plato View Post
    $\displaystyle \sqrt{n^4 + 5n + 10}\le\sqrt{n^4 + 5n^4 + 10n^4}=4n^2$
    How? I'm sorry I don't understand how you get 4n^2
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  8. #8
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    Re: Determine whether the series converges or diverges.

    Quote Originally Posted by moonman View Post
    How? I'm sorry I don't understand how you get 4n^2
    $\displaystyle 4n^4+5n^4+10n^4=16n^4$

    What is the square root of that?
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  9. #9
    Junior Member moonman's Avatar
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    Re: Determine whether the series converges or diverges.

    Oh gosh! Thanks, I feel like an idiot now. I did not follow the order of operations.
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