Series Convergence/Divergence tests: (arctan n)/(n^2 + 1)

[Jamers328]

Determine if the following series converges or diverges. Which test did you use?

The series (arctan n)/(n^2 +1)

We have to use tests like the limit comparison test, the p-test, the comparison test, and the integral test.

 

[stapel]

Which test did you use?

You use whichever test is successful for you! :wink:

What have you tried so far? What were the results? What are your thoughts? :?:

Please be complete. Thank you!

Eliz.

 

[pka]

Note that the arctangent function by \(\displaystyle \frac {\pi} {2}\).

 

[Jamers328]

Sorry Eliz. I tried the Comparsion test, and it didn't work... I used it with 1/n though, I am not sure if that's correct. I tried the Integral test, and didn't get anywhere really... maybe I just wasn't doing the integral correctly, I'm not sure. I just need a start... a good test to use, and what series to compare it to if necessary.


Thanks pka!

 

[Jamers328]

I used the integral test. Thanks for your help guys.

 

[pka]

I tried the Comparsion test, and it didn't work.

Why not?
The series \(\displaystyle \sum {\frac{1}{{n^2 + 1}}}\) converges by the p=2 test.
Thus \(\displaystyle \sum {\frac{{\arctan (n)}}{{n^2 + 1}}} \le \frac{\pi }{2}\sum {\frac{1}{{n^2 + 1}}}\) you see the series converges by simple comparison.