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Math Help - Series Convergences....

  1. #1
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    Series Convergences....

    This is a simple series, it was listed in the review section of my book, they state for this problem determine if the series converges or diverges by using the integral test, however, I feel limiting myself to one method of showing convergence will not help me on the test, so I need someone to check my different ways of showing this converges

    The initial problem \sum_{n=1}^{\infty}\frac{1}{(n+2)^2}

    Using the integral test

    \sum_{n=1}^{\infty}\frac{1}{(n+2)^2} = \lim_{b\to\infty}\int_1^b \frac{1}{(x+2)^2} dx

    Using some simple substitution to find the integrand of
    <br />
\lim_{b\to\infty}- \frac{1}{(x+2)} |_1^b

    Then

    <br />
\lim_{b\to\infty}- \frac{1}{(b+2)} + \lim_{b\to\infty}\frac{1}{(3)}=\frac{1}{3}

    Since the limit is finite the series converges.



    Using a direct comparison

    \sum_{n=1}^{\infty}\frac{1}{(n+2)^2}\le\sum_{n=1}^  {\infty}\frac{1}{n^2}

    We know by the p-series test for p>1 converges, thus the series converges

    We can also take the limit of the series itself and apply L'Hopital's rule and see the limit is 0 but this doesn't mean much, because it could also diverge or converge
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  2. #2
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    Krizalid's Avatar
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    definitely, the best and faster way, it's the comparison test.
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  3. #3
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    It seems so in the cases your can use it. I am sure you can use it in any case but my skills are not that great, I work with what I can and improve with I have little by little. It seems in most scenarios the integral test is recommend with fraction with a higher power in the denomiator
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