1. ## 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.

2. 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.

3. 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