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Math Help - series convergence

  1. #1
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    series convergence

    does the series: summation from 1 to infinity (-1)^n (n^2) / [n^2 + n] converge?

    why cant i say that
    since (n^2) / [n^2 + n] is decreasing and greater than 0, by alternating series test, it converges?

    the ans says that it diverges.
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    does the series: summation from 1 to infinity (-1)^n (n^2) / [n^2 + n] converge?

    why cant i say that
    since (n^2) / [n^2 + n] is decreasing and greater than 0, by alternating series test, it converges?

    the ans says that it diverges.
    Note that

    (-1)^n\frac{n^2+n}{n^2}=(-1)^n\left(1+\frac{1}{n} \right)

    This limit does not go to zero (it does not exist) so the series diverges by the basic divergence test.
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  3. #3
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    Quote Originally Posted by alexandrabel90 View Post
    does the series: summation from 1 to infinity (-1)^n (n^2) / [n^2 + n] converge?

    why cant i say that
    since (n^2) / [n^2 + n] is decreasing and greater than 0, by alternating series test, it converges?

    the ans says that it diverges.

    Does \lim_{n\to\infty}\frac{(-1)^nn^2}{n^2+n}=0 ? Sometimes it is highly advisable to think a little about a problem BEFORE rushing up

    to ask for help...

    Tonio
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