Question

Direct Comparison Test/ Limit Comparison Tests

Use one of these tests to find if they are divergent or convergent

I can actually find by integrating these improper integrals but I'm supposed to use one of these tests.
It would be great if the limit comparison test was used because I don't know how to choose the function for the comparison.
Explanations please!
Questions:

Answer 1

the choice of function is more intuitive than anything

Answer 2

\[\int\limits\limits_{\pi}^{\infty} (1-\sin(x))/x^2\]
\[\int\limits_{2}^{\infty} xdx/\sqrt{x^4-1}\]

Answer 3

I chose \[x/\sqrt{x^4} \]
and \[1/\sqrt{x^4}\]
but it ended up getting simplfied and ended up a hard to solve integral or ended up with divergent which is not true.
The first one, I have zero idea on what to choose.

There are other problems but I hope if I understand this, I can solve them by myself.

Answer 4

\[\frac{1-sin(n)}{n^2}\]
for large values of n that might resemble \(\frac{-sin(n)}{n^2}\) which really looks to mimic \(-\frac{1}{n^2}\)

but it might be good to get a view of it on the wolf first

Answer 5

http://www.wolframalpha.com/input/?i=y%3D%281-sin%28x%29%29%2Fx%5E2+and+y%3D-1%2Fx%5E2

Answer 6

It might sound rude but I really don't understand the point of these tests when it might be simpler to just integrate them. Arrrgh! >:(

Answer 7

the point is just for gaining a bit of experience .... some hands on work if you will :)

Answer 8

http://www.wolframalpha.com/input/?i=y%3D%281-sin%28x%29%29%2Fx%5E2+and+y%3D-sin%28x%29%2Fx%5E2

Answer 9

im not to adept at these comparisons either ...

Answer 10

:/