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Math Help - Inconclusive

  1. #1
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    Inconclusive

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

    I tried the ratio test and I got 1, which means inconclusive. What can I do to tell whether it is converging or diverging?
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  2. #2
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    Quote Originally Posted by fastcarslaugh View Post
    \sum_{n=1}^\infty (-1)^{n-1}\frac{\sqrt{n}}{ n+2 }

    I tried the ratio test and I got 1, which means inconclusive. What can I do to tell whether it is converging or diverging?
    nth term goes to zero ... terms of the series alternate in sign ... what does that tell you?
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by fastcarslaugh View Post
    \sum_{n=1}^\infty (-1)^{n-1}\frac{\sqrt{n}}{ n+2 }

    I tried the ratio test and I got 1, which means inconclusive. What can I do to tell whether it is converging or diverging?
    Let a_n=\frac{\sqrt{n}}{n+2}. Note that \forall{n}\in\mathbb{N}~a_{n+1}\leqslant{a_n}. Further note that \lim_{n\to\infty}a_n=0. So this series converges by Leibniz's Criterion. Note that although the Ratio Test is very useful, a lot of kids get in a Ratio Test only mindframe, try not to do that.
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    \sum_{n=1}^\infty (-1)^n\frac{n}{ 5 + \ln n }

    so this is also alternating. and it looks like it's oscillating. Therefore, it's diverging?

    Also
    \sum_{n=1}^\infty (-1)^n\frac{ 4 n }{ 8 n + 7 }

    This one is oscillating. it means diverging, right?
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by fastcarslaugh View Post
    \sum_{n=1}^\infty (-1)^n\frac{n}{ 5 + \ln n }

    so this is also alternating. and it looks like it's oscillating. Therefore, it's diverging?

    Also
    \sum_{n=1}^\infty (-1)^n\frac{ 4 n }{ 8 n + 7 }

    This one is oscillating. it means diverging, right?
    In this case oscillating and alternating are synonomous. And secondly no that is completely incorrect. Holistically being alternating greatly increases the chances a series converges. It mus satisfy much less stringent requirements. Do you know the Alternating Series Test (Leibniz's Criterion)?
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  6. #6
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    Quote Originally Posted by Mathstud28 View Post
    In this case oscillating and alternating are synonomous. And secondly no that is completely incorrect. Holistically being alternating greatly increases the chances a series converges. It mus satisfy much less stringent requirements. Do you know the Alternating Series Test (Leibniz's Criterion)?

    No, Could you explain it?


    <br />
\sum_{n=1}^\infty (-1)^n\frac{n}{ 5 + \ln n }<br />

    Although this seems to bounce back and forth in sign, it looks like it's approaching a certain value.
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  7. #7
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by fastcarslaugh View Post
    No, Could you explain it?
    The Alternating Series Test (Lebniz's Criterion) loosely states: If \sum{(-1)^na_n} is an infinite series it converges iff \lim_{n\to\infty}a_n=0 and a_n\in\downarrow or if you prefer a_{n+1}\leqslant{a_n}
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  8. #8
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    Pauls Online Notes : Calculus II - Alternating Series Test

    you're not an "online" student, are you?
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  9. #9
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    Quote Originally Posted by fastcarslaugh View Post
    \sum_{n=1}^\infty (-1)^{n-1}\frac{\sqrt{n}}{ n+2 }

    I tried the ratio test and I got 1, which means inconclusive. What can I do to tell whether it is converging or diverging?
    Try the alternating series test for conditional convergence.

    Never mind way to late I guess
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