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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Originally Posted by alexandrabel90 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 $\displaystyle (-1)^n\frac{n^2+n}{n^2}=(-1)^n\left(1+\frac{1}{n} \right)$ This limit does not go to zero (it does not exist) so the series diverges by the basic divergence test.
Originally Posted by alexandrabel90 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 $\displaystyle \lim_{n\to\infty}\frac{(-1)^nn^2}{n^2+n}=0$ ? Sometimes it is highly advisable to think a little about a problem BEFORE rushing up to ask for help... Tonio
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