sum n=1 oo (arctan(n))^2 / 1+n^2
how do i solve it using integral test?
Less formally than Jhevon, as $\displaystyle \arctan(n)$ by definition is giving a value in the range $\displaystyle (-\pi/2, \pi/2)$ $\displaystyle \arctan(n)$ is bounded (in fact not only is it bounded but as $\displaystyle n \to \infty \arctan(n) \to \pi/2$). Also for large $\displaystyle n$, $\displaystyle 1/(1+n^2)$ behaves like $\displaystyle 1/n^2$ so the series is absolutly convergent.
RonL
the comparison test. he wasn't being very formal though, since i already answered the question.
The comparison test says:
Let $\displaystyle \sum a_{n}$ be a series where $\displaystyle a_{n} \geq 0 $ for all $\displaystyle n$.
(i) If $\displaystyle \sum a_{n}$ converges and |$\displaystyle b_{n}$|$\displaystyle \leq a_{n}$ for all $\displaystyle n$, then $\displaystyle \sum b_{n}$ converges
(ii) If $\displaystyle \sum a_{n} = + \infty$ and $\displaystyle b_{n} \geq a_{n}$ for all $\displaystyle n$, then $\displaystyle \sum b_{n} = + \infty$.
CaptainBlack used the fact that:
$\displaystyle \sum \left| \frac {\arctan^2 n}{1 + n^2} \right| \leq \sum \frac {(\frac {\pi}{2})^2}{1 + n^2} = \frac {\pi^2}{4} \sum \frac {1}{1 + n^2}$ and $\displaystyle \sum \frac {1}{1 + n^2} $ converges by the comparison test, if we take $\displaystyle \sum a_{n}$ to be $\displaystyle \sum \frac {1}{n^2} $ and $\displaystyle \sum b_{n}$ to be $\displaystyle \sum \frac {1}{1 + n^2} $
EDIT: Bare with me, i'm just learning LaTex, it will take me a while to get used to it
EDIT 2: Finally! It is ready!
EDIT 3: I'm so proud of myself ...even though typing that took forever. i guess i'll get faster with practice though, and after i memorize the commands
None explicitly, but knowledge of the behaviour of series. This is an informal
argument of the form that we actually use if we want to know if this thing
converges, only later will a formal proof be constructed to make the argument rigorous.
The purpose of such an argument is to explain why this converges, which is not always
clear from a formal proof (Sound of Bourbaki turning in his grave from off stage right).
Jhevon has explained what the idea is in more detail
RonL